Someone has twisted the adjustment screw on the neutralizing capacitor C706.
This is a square white-ceramic body part bolted to the underside of the chassis, about an inch square. It's between the driver and final tubes. Has "304M" printed on one side.
Its purpose is to cancel out the unwanted RF feedback that takes place inside the final tube.
But the adjustment is a bit finicky. What usually happens is that someone twists the adjustment screw while watching the wattmeter, hoping for just a little more on the meter.
This will totally disrupt that neutralizing circuit, and cause the final tube to become an oscillator on sideband transmit. The frequency that it now sends out to the antenna is not anywhere near the channel that's selected. If you had a frequency counter in line with the coax, you would see an unstable reading, probably around 28 MHz.
Simple fix is to simply crank that adjustment screw tighter. Odds are that it's fairly loose now.
Any time you adust C706 you will also have to peak the Plate Tune on the rear panel, and the driver coil T700. Any change to the setting of C706 will move the peak setting of those two adjustments, and require you to repeak them.
I'll skip the full adjustment procedure for C706 for now. If simply tightening it stops the SSB carrier problem, that's worth trying first.
Just be sure to re-peak T700 and the Plate Tune control on the rear panel any time the setting of C706 is changed.
Oh, and the hum on SSB receive may be from the 14-Volt DC power that comes from the plug-in "BA" board next to the audio tube. The filter capacitors on that board will certainly be bad if they are original.
Might be from a bad tube.
73