I've already helped him out by identifying the problem while you continue to confuse him into thinking he needs a new meter or antenna. It takes years of experimenting with solid state amplifier design before you gain the experience others like Bob have shared on this topic.
Crusher is right to be skeptical blaming the meter while we only have a two transistor 12 volt amp in front of it. The meter obviously has a scale capable of reading 250 watts from his description. I think we can all agree this pair of transistors would struggle to reach that level in PEP watts much less with a 2 watt carrier as drive.
Yes, the diodes in some cheaper meters can have their junctions saturated from too much RF but once you understand the symptoms here, the meter is no longer suspect. There are very few things that will make the VSWR appear higher under carrier conditions than they are with modulation. Downward swing or an unstable amp make up the short list.
With only 1 watt of carrier drive per transistor and many times that with positive audio peaks, we can check downward swing off the list quick. His VSWR is good with no amp and not so bad when the amp is driven with modulation. It's at its worst someplace in the middle under low drive conditions. Exactly when an unstable amp is most likely to act up.
Furthermore, the Palomar type solid state design has a long history of this specific problem dating back decades. Several threads have already been written on this forum covering this topic extensively. Right down to how to add the missing negative feedback and tuned input circuits.
PS: If it's what I suspect there is a very strong chance this amplifier will perform horrible on SSB where the drive level is continuously transitioning between low and high levels. One of the first things you should check is that the watt meter on SSB is behaving like a modulation meter on AM. If the watt meter is not in step with your audio peaks and fails to decrease quickly between words, you've further confirmed oscillations are present.