Pondering a trip to Ireland next summer. How are the crowds?

Summer is indeed prime time for visiting Ireland and you'll experience some crowding at any of the more popular tourist destinations. It's typically not a problem however, even if you mistime your arrival at a particular place just as a tour bus has disgorged its load - you'll just have plenty of company for a few minutes. The work-around is to start early - before the tour buses start to roll around 9 AM, or later in the afternoon (after around 4 PM) when they're finishing up for the day. In between you'll generally find that heading out into the countryside and off the beaten tourist path will be pleasant and relatively uncrowded.
Know that summer days are quite long in Ireland so you'll have plenty of daylight to work with as you plan your days.
Renting a car would provide maximum flexibily for excursions into the countryside - you won't need (or want) one in the cities.
Of the places you've mentioned our personal favorite is Donegal, but then we've been to Ireland quite a lot and have seen most of the major tourist sites.

We were there in July. The core area in Dublin was crowded in evenings, with many young Americans at the pubs and restaurants. And the streets of Dingle were also busy. But it was manageable. I think you can always find a time or place to get away from crowds, but it's worth it regardless. We took a DART ride out to Howth (a suburb or Dublin on the sea) and it was a very nice day trip. Ennis is nice and quiet, but not a lot to see or do there other than shops and restaurants. It does have a non-touristy feel to it. Kilkenny as well.

We didn't think it was overly expensive. Wherever you're traveling on vacation it seems like you're hemorrhaging money just for everyday expenses.

Outside of Dublin, accommodation is fairly affordable. Once exception, IME, is Derry. I was surprised how difficult it was to find an affordable place to stay there. And I was disappointed by the restaurants in Derry. I don't regret visiting, though.
The car rental seems to be very expensive, while public transportation is very affordable. In the West, it's mainly buses. If you're comfortable with the "wrong side" driving, then by all means, get a car. It gives you the flexibility to go wherever you want and to stay in places that maybe don't get a lot of tourists.
County Donegal is a good choice, as it is home to incredible scenery, including the Slieve League (you can see it from land and sea, easily done from the lovely Killybegs). I'm not familiar with Armagh (city or county), which probably suggests that it's not a terribly crowded place. The best way to avoid crowds is to avoid Killarney, Kenmare, Dingle, Doolin and Galway, IMO. The coastal areas north of Galway (and especially north of Westport) are less touristed, as are the interior areas.
I don't have much experience with Scotland, but I expect it's similar in that Edinburgh is crazy expensive like Dublin, and smaller cities/towns/villages are more affordable. You can travel between the two by plane or by ferry (don't take the car over, though; too expensive).

Dumfries certainly doesn't get crowded with tourists, and the pricing of lodgings is pretty modest, certainly compared to most of the Highlands, and you categorically do not need to be booking far, far ahead of time (either for hotels or restaurants).
The same applies to pretty much the whole of the county of Dumfries and Galloway.

You don't say if you are thinking about Dumfries for Rabbie Burns, John Paul Jones or for other reasons.

It depends where you intend to visit as the miles along the A75 are long, so it may pay to be based in one of the smaller towns further west.