First you need to figure out "how is water getting into my wall"? Do you have access on the other side. Is there an apparent gap?
If you can push tile, there is literally mush/mold behind it. Your solution is simple. Take down tile and wall and fix it right.
There is no other solution because the all involve more money, more time, and more mold in your home. If you want to go super cheap temporarily then affix a shower curtain to that wall so that it sees no water.
Redoing a wall/tile in a tub surround is light years easier than a shower. It can be demo'ed with first layer and waterproofing in one day. And honestly if you just need cheap then by a tub surround kit and put up the PVC surrounds directly on the studs. Cheaper is buy these kits from a local habitat store.
And no you don't take half the wall down. At the very least the faucet wall and side wall have to come down. You are talking to an ex flipper that would do things in the least costly way - there is no saving money when there is a water problem. Go full force fix it and then there aren't add-ons and gotchas. There is nothing that should be half done in your shower if it is that bad.
Also not sure about COVID prices but an already framed tub demo plus basic tile install has put me back about $1000-1500 (1500 was some extras). Materials - drywall, concrete board, waterproofing, decent/cheap tiles... $6-800... and again if you go to a habitat store you could find boxes of tiles that will work for like $2 a tile... do the math.