New speakers, what Amp do I need?

This does not compute.

 

The cheapest fix would probably be going back over the speaker installation and adding a bit of glue and insulation here and there to reduce rattles and sympathetic vibration. Your speakers are probably making great sound, but they may also be causing a part of the door to vibrate almost as loud and with far less control, which muddies up the sound badly.

The loudest fix would be adding a 4-channel amplifier, something north of 50W per channel. You would use one channel per door. The front door speaker would be wired in parallel with the same-side tweeter. (You are probably already wired like that for the JVC)

The bump-est fix would be adding a powered subwoofer. That way the door speakers don't have to work nearly as hard to produce bass sounds- they aren't very good at it anyway. A sub has the physical size to produce those sounds easily, and with a built-in amp it's relatively easy to install. This removes much of the workload from the JVC built-in amps, so they will sound much better driving the new RFs.

 

I think it is even more likely that the distortion is from having speakers of the wrong impedance for the amp in the radio.

I suggest a visit to a respected "custom car audio" shop in the area.