I have never been to Albania, but I've traveled around the Balkans a fair amount.
My rule of thumb concerning buses, and most especially buses in the Balkans, is this: The only schedule information you should totally rely on is what you obtain at the bus station in the departure city.
My second rule of thumb: International buses are relatively uncommon. They frequently start the run full or nearly so. Buy your ticket as soon as you know which bus you want to take and can figure out a way to make the purchase.
The other obvious stopover possibility besides Kotor is Budva. Kotor is probably more prudent, because there are direct buses from Kotor to Dubrovnik and I'm not sure you can travel from Budva to Dubrovnik without changing buses in Kotor. You will note that the bus also stops in Podgorica, the capital of Montenegro. It's not a particularly attractive place. Both Budva and Kotor have nice historic districts.
I would be uncomfortable depending on one specific bus to get me to Dubrovnik in time for a flight, so I hope you are planning to spend a couple of days there. That will allow some recovery time if there is an issue with the bus you intend to take.
Montenegro uses the euro. You will probably find it difficult to spend leftover Albanian leks at the bus comfort stop(s) after you cross the border. Either hit a currency-exchange booth (check carefully to be sure the rate is decent and fees are modest) to buy a few euros in Albania or take plenty of refreshments with you. You may encounter a toilet attendant you are expected to tip; he or she may accept other currencies (I don't know about leks; they aren't very useful), but it will be at a very, very poor exchange rate, and you can expect to get some attitude about it.
I don't know the policy on Albanian buses, but Croatian buses and I think also Montenegrin buses usually charge a fee of about 1 euro (or equivalent) for each bag stored in the hold.