Fixing shower tile to repair possible water damage

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.

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?