My (single ended, class A) tube amp sounds great while it's shutting off. What's happening?
Measuring B+ voltage is a good idea. A class A amplifier draws full plate current continuously, so it will drain the power supply capacitors quite quickly. The heater cools down slowly, so I think it is the B+ voltage drop that is causing most of the compression effect. 

One characteristic of Pentodes is that the Plate draws relatively constant current at different voltages (unlike a Triode, which acts like a resistor). As a result the operating point will shift off center when the voltage is lowered, causing compression of the negative half of the output waveform. 

In the curve trace below I have added load lines for normal B+ voltage (red) and reduced B+ voltage (blue) to show what I think might be happening. For lowest distortion the distances between Grid voltage steps along the load line should be equal. In the blue line you can see that the steps are compressed at the lower voltage end, and because the operating point (vertical blue line) is closer to that end the distortion is higher.      



Once you have determined what voltage produces the desired sound the next problem is how to get it there in normal operation. Tube amps often use a two stage power supply filter with a resistor between the first and second capacitors. You might be able to lower the B+ voltage by increasing the value of this resistor.



Running the heater at lower voltage is not a good idea because the Cathode coating could get 'poisoned' (and it probably won't produce the sound you want anyway, since the lower emission will probably just reduce the gain and make it quieter).