We have been to Yosemite quite a few times over the years. If there is one place to relax your budget, it's Yosemite. My first choice would be Yosemite Valley Lodge. It's not fancy (but I believe that they remodeled a few years ago). The location is perfect - right in Yosemite Valley ("the Valley"). You can park your car and use the shuttle to get around. There are openings in August, although it looks like it is filling up fast, and the rates are about $275 per night. This is 100% where I would stay.
https://www.travelyosemite.com/lodging/yosemite-valley-lodge/ My second choice would be to stay at a vacation home in Yosemite West, about a half hour from the valley. I have used this property manager before
https://www.scenicwonders.com/yosem...MI0bChrIvx5AIVRR6tBh2KcQOrEAAYAyAAEgLDkvD_BwE, but I am sure that there are listings on AirBnB and HomeAway. There is a small condo development in Yosemite West (inside the park - so no going through the gate every day) and those will be in your price range. Personally, I think that any other areas (El Portal, Oakhurst, Wawona, etc) are too far away for such a short trip. If it were me, I would bite the bullet and pay whatever you have to to stay in the Valley.
August could be pretty warm (80's - 90's maybe?), and none of the lodging that I am aware of is air conditioned. It just is what it is. Yosemite West is at 6000' elevation vs the Valley at about 4000'. But it gets nice and cool in the evening in both places. We usually camp, but have stayed in both the Lodge and in some of the homes in Yosemite West when it's too cold to camp.
And while I agree that Wawona is delightful (and the historic hotel there is so charming) it's a 50 min drive into the Valley. For a two - three day visit, I would do whatever I could to minimize the time spent in the car.