3-5 nights, what should we do?!

I would do option 1. Dubrovnik does indeed suffer from overtourism, but I don't think it should be missed when in that region. I was there for four nights on a thorough tour of Croatia last September. We did not visit Split, though, so I have no opinion about that town. One way to mitigate cruise ship impact on your visit would be to stay in the Lapad area, rather than in the old town itself, and do the bulk of your old town sightseeing in the earlier morning or late afternoon and evening, when the cruisers have not yet disembarked, or are back on their ships for dinner and departure, and it's less overwhelmingly hot and crowded. Lapad is a short bus ride from the old town, has a variety of hotels and restaurants (our group stayed at Hotel Kompas, which is very nice), a scenic waterfront promenade, and quite a good little beach. A daytrip to Montenegro is a good thing to do, but you won't necessarily escape cruise ships by doing that, as your likely primary destination, Kotor, is a smaller version of Dubrovnik that is also a cruise ship port call.

Since you are so young you can definitely visit both cities. The ferry from Dubrovnik to Split is 4.5 hours but is a very pleasant ride. Just don’t use Jadrolinija.
I would stay in the old town to minimize travel in and out of the city. We’ve been there over a half-dozen times and never stay outside the wall. Especially if you are going to split your time. Staying inside the wall gives you time to walk around at night when day trippers are gone and not worry about the bus. FYI, the buses do not run too late at night but cabs are inexpensive if you stay in Lapad.
On a crowded cruise ship day you can take the 15 minute ferry to Lokrum island. The ferry leaves from the port in old town. There is a very nice, never crowded, exhibition on GofT and its filming. You will recognize many sights from the movie. There is even a replica throne you can sit on - no charge for the exhibition. There is a lovely cafe on the island, only one, run by the same family for years. You can also do some hiking on the island or go swimming. We usually spend the entire day there.
Also on cruise ship day, get on the wall first thing in the morning, you’ll miss the crowds.
Also with regard to Split, stay within Diocletian’s palace. We always stay at the Peristil Hotel. Family run and breakfast included. You walk out of the door right into the middle of the city.
We visit yearly because of family, but both Dubrovnik and Split are what I would call a ‘one and done’ city. For most people, one visit is enough.

Any Rome flights from Podgorica?

3-4 nights in Dubrovnik is perfect. Yes, it does totally get crowded (and moreover HOT) during the day. But it can be pretty easy to avoid the worst of this by just taking on an activity that keeps you out of the old town at peak hours. Beach/swimming is the order of the day here, and you have good options. The trip out to Lokrum is nice. I’d also recommend taking the boat to nearby Cavtat, which compared to enclosed Dubrovnik was breezy and cool. Some really nice swimming areas with beach bars just north of town. It’s a genteel break from the madness. There’s also tons of half-full day excursions by boat or kayak. By the evening, a good chunk of the crowd is back to their big hotels or cruise ships. Still mobbed of course, but the town is built for it. I don’t quite know how, but Dubrovnik manages to mostly be jammed with tourists and still be a blast.

I’d still try to stay in the old town, and book it now. Having the ability to quickly duck into your room for a break or to freshen up versus trekking back to Lapad or up the hills is worth the expense imo.

Also, FWIW I also loved Split and think it’s a fine choice if Dubrovnik looks too daunting.

There does appear to be one direct flight between Podgorica and Rome--Google Flights says "flex flight, no price available" on Air Montenegro. I'll investigate further but I really appreciate all of these suggestions! Re: cruise ships in Kotor, on the day with 4 ships in DBV there is only 1 in Kotor so we thought that might be a good option on that day. It's nice to hear that DBV handles the crowds well overall though!

Unsolicited thoughts here: You don’t say where in Bosnia you are going but 4 nights in Sarajevo and 2 nights in Mostar would not be too many for me. That leaves you 3 for Dubrovnik - which would also be about right for me. And I agree with Dubrovnik over Split.

But if you find the good flight from Podgorica, a leisurely drive of 3 nights through Montenegro would be gorgeous! The flight from Podgorica to Rome is only twice a week, late afternoon, and economy in August is about €60. It looks like that includes a piece of checked luggage plus a carry on of 8k. And it’s about a 4 hr drive from Mostar. But you could make a gorgeous stop of 2-3 nights - either mountains or coast, your choice - on the way. I can give you the name of someone who could arrange a driver for not much money.

TexasTravelMom, certainly not unsolicited--I read your recent trip report and really enjoyed it! As of now we have 3 nights in Sarajevo and 2 in Mostar planned, but nothing is booked yet in terms of accommodations or whatnot so everything can be changed other than our flights. I'm torn because I would hate to miss Dubrovnik but Montenegro does seem really lovely! I wonder if we cut a night from Mostar and did 2 nights in Dubrovnik and 2 in Montenegro if that would be totally crazy... we do have a full 9 nights in Rome so I feel like we can afford to bounce around a bit more the first half of the trip but I don't want it to be a total blur.

Hmmm, no bad choices. BUT, you say you are moving to Rome in 2025, which means that getting to and from some of these countries is going to be easier. With the difficulty of flights to Rome from Sarajevo, I might advise spending 6 nights in Bosnia and then 3 nights either on your way to Podgorica or 3 nights in Dubrovnik. You can already tell it will be easier to get back to either Croatia or Montenegro and more difficult to get back to Bosnia. So invest an extra night there.

Great point!

Of the options you listed, I'd lean toward only Dubrovnik. But I would be inclined to save that for another trip with Split, Kotor, Zadar, and the islands, or at least outside of high season.

I'd probably go the other direction, and fly to Rome from Belgrade. Mostar could then be a day trip from Sarajevo. Belgrade will also be a little less scorching hot in August.

As of now we have 3 nights in Sarajevo and 2 in Mostar planned, but nothing is booked yet in terms of accommodations or whatnot so everything can be changed other than our flights. I'm torn because I would hate to miss Dubrovnik but Montenegro does seem really lovely! I wonder if we cut a night from Mostar and did 2 nights in Dubrovnik and 2 in Montenegro if that would be totally crazy... we do have a full 9 nights in Rome so I feel like we can afford to bounce around a bit more the first half of the trip but I don't want it to be a total blur.

Two nights in Sarajevo is a lot better than missing it. It will either speak to you and you will want to return or you will wonder why you went there in the first place. Personally, it spoke to me.

Two or three nights in Dubrovnik is all the Disneyland I could endure. Stunning old town in the evenings though. Glad I went and think its well worth 2 full days.

I havent been to Mostar. I hear it s a 1 or 2 night stop.

Montenegro is one of my favorite places. But its a week trip on its own. But a night in Perast on the bay with a stop and a tour in Herceg Novi on the way up or back is well worth the effort. Best with a guide and a car (but that is cheap in that part of the world). You having seen Dubrovnik will probably find Kotor a bit of a let down, so its more than okay to skip it despite it being lovely in its own way.