Might swollen caps be working properly?
Replacing the caps is a good idea.

The deformation is caused by a high pressure within the cap, which means that the electrolyte (liquid!) became so hot that some of it vaporized. For this to happen, you need a failure mechanism like (i) excessive leakage current, caused by overvoltage or a failure of the dielectric oxide layer on the aluminum foils or (ii) an increased internal resistance of the caps, caused by an aging electrolyte. Note that once some of the electrolyte vaporizes, the resistance will increase further, making the effect self-accelerating until the capacitor has lost all of its electrolyte and becomes "dead" - it is physically still present, but won't act like much of a cap any more.

The deformation also increases the risk of the cap developing a short circuit, which might cause further damage to the "upstream" parts of the system (power supply, transoformer, rectifier, ...).