My humble opinion:
1. You can't play Padel w/o going to gym; well you can but eventually you're gonna get injured. Need to build those muscles to amortize extreme loads and hard surface. You say you are doing workouts so those should help, but since they are not, find/change coach and ask for guidance specifically targetting padel (strenghening back chain of legs, beef quad for amortization, ankle and calfes, shoulders and arms, etc). Probably your workouts are not effective, i.e. you don't give enough load for muscles so they could grow, that's something good coach can help out with. 3 times a week is minimal attendance that can make you better, for 2 you'll probably get stuck at some level
2. Go to your doctor/endocrinologist and check out your blood results; see what's lacking or off the charts, fix it.
3. Solve for protein intake, check your food intake and make sure that you get enough proteins. Around 40 and further you'll need more of it and less carbs/sugar. There's a lot of guides about it on the internetz.
4. Sleep is important, need to have stable 7-8 hours of it, ideally without wakeups/interruptions during the night. r/sleephackers r/sleep to help:)
5. Folks mentioning it here but yes, padel court is a hard and rough surface, especially if you play in some old warehouse remade for padel. While you playing you stress out your ligaments and joints along with the muscles, the biggest load would be related to joint amortizations because of surface impacts. At some moment your body would just run out of the capacity, and you will start to damage yourself. Shoes for sure can help, but golden rule of the thumb -> if you want to play long, don't overuse and make sure you recover b/w sessions. For me it's at least 1 day b/w games.