Looking at most of your photos I have been thinking that the camera may be part of the sharpness problem.
If you're looking to upgrade I would buy into a system you could grow into. Like a DSLR with interchangeable lenses. All I have ever shot is Canon so that's the only system I feel like giving specific recommendations on.
I think Nikon is a better value for your money (get a little more bang for your buck). Sony is a good choice too. I keep forgetting which Nikon
@Alyssa-Bee uses but it's a good value setup that can produce great results.
The hardest part is finding a lens that performs well at a low price point.
If you go with Canon, the best lens for the money is their EF-S 55–250mm f/4–5.6 IS STM lens. It is a very sharp lens and can be bought used for less than $100. The only downside to this lens is 250mm is a little short on the long end. I used mine for a while but wanted more reach so I ended up buying a 400mm which was very expensive. The 250 will work you just have to practice getting a little bit closer to your subjects. But for your budget I think it is the best choice.
As far as bodies go, I pretty much always recommend the Canon 80D. Not just because I owned one but because it is a good balance between price and performance. You can find them used for between $400-500. You definitely will want a canon with the dual pixel autofocus. I believe that started with the 70D. You can pick up a Canon rebel t7i (make sure it has the "i") for a little bit cheaper and it has the same sensor as the 80D. It is basically 90% the same as an 80D, but if you want a camera you can grow into I would go with the 80D. It has weather proofing, a better button layout, and a few extra features like micro focus adjustment that you can tune to each lens.
Regardless of which brand you choose I definitely recommend going with a DSLR and buying used.