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.