Fixing shower tile to repair possible water damage

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.

am I just hiding the problem if I regrout the tiles?

If the wall behind the tiles seems spongy i.e. "flex a bit", then yes you probably have a bigger problem in that the substrate (the wall material behind the tiles) may be wet.

Will I get worse water damage down the line?

Whatever is causing the damage you have now will continue to cause additional damage if you don't fix it.

Is regrouting the tiles myself a bad idea?

No, but not fixing the root cause of the damage is a bad idea. With a bit of research and carefully following instructions you are certainly capable of all parts of this repair, including the final step of applying grout. The first step is identifying the cause of the damage and fixing that.

Your picture does not seem to show any terrible problem with either the tile or the grout, so it's tough to help you with just the information provided; however, a big sign that you do mention is spongy wall behind tile. You most certainly need to open that wall up to inspect, maybe from the other side first to see what there is to see?

If your tiles are flexing its likely that the wall behind has deteriorated because it became wet. Its possible that it was not installed from either moisture resistant drywall or from cement board and that the silicone seal has deteriorated. You may have water seeping behind your tiles leading to mould growth.

Re-grouting and re-sealing are your cheap, but temporary solution. If those tiles are flexing, in 2-3 years the silicone will break away again and deterioration will continue.

Base this decision on your current home budget. If you have the money to do a complete replacement without getting into debt, do it. You have to get it done anyway. If you do not, re-grout and re-seal and save up for a complete replacement.

Consider that if you do a complete tile replacement (as you should I think), your older bath may look out of place and your fixtures may look out of place too.