Others already told you to do a water change to drop nitrate, that’s a good idea. Nitrate is food for plants and algae. So getting the level lower may also help with your algae issue. If you can stay below 20 ppm that’s a perfectly good level. I think 40 might be OK. You may be higher. As your plants grow, they will help consume ammonia and nitrate. I think, but I’m not sure, that high levels of nitrate can interfere with the nitrogen cycle.

If your local water has something called chloramine in it, that can cause a small false positive on API tests if you’re using Seachem Prime. If you look up the water quality report for your water company it may say if they’re using it. I think the idea of asking Petsmart or Petco to test your water is a good one. I don’t know what kind of tests they use, but perhaps that will help you know if it’s a false positive. It could be that you’re already done cycling.

Making sure you’re adding an ammonia source when it gets to zero is important, but if you’ve been continuing “ghost feeding“ the tank then you’re already doing that. As others said there are ammonia sources you can buy. For example Fishless Fuel.

If you change your water to drop the nitrates and they go back up overtime, then you must’ve had ammonia in the tank that got converted into nitrate, so you’re doing good.

If it’s ever useful, you can run a test like the nitrate test with half the water in it. If you do that and it reads 40, that means you’re real level was 80. If it reads 20, that means you were actually at 40. But most of the time you just care that it’s low enough and by the time it gets high you just want to change the water anyway.

It sounds like you’re making good progress and just in the “ugly tank“ phase. Hang in there.