A 60 cycle hum (just slightly higher in pitch than the "A" string of a bass) could be feeding in from the power supply for the reverb recovery amp. This power supply only features 1/2 wave rectification which would give a 60 cycle tone just waiting to be injected into the signal, and it would be louder when the reverb control is zero... and not dependent on volume control settings. The first thing I would do is replace the two 1000mfd/35 volt caps (locate point "D" in the power supply section of the schematic located on this site -
http://www.sunnamplifiers.com/schematics/sceptre.gif )
If you can find larger than 1000mfd that won't have trouble physically fitting in the chassis, by all means use them... this will increase the filtering ability which is real important in 1/2 wave rectifiers.
What kind of noise are you hearing with the footswitch? The footswitch connection inside the head goes to the input of the reverb recovery transistor... along with the output of the reverb tank. If the noise is scratchy sounding, then you could have a bad solder joint in the switch, the switch could need replacing, or the switch jack in the head could be dirty. I would clean the switch jack in the head first, if it has been unused for some time, then it can build up crud that won't make a good contact. If you cannot switch off the reverb with the switch, then I would suspect the switch or a solder joint at the switch terminals.
These older amps won't have a lot of classic tube overdrive sound - the circuit is no where near what creates what most people associate with classic tube overdrive. The power amp is a hi-fi ultralinear design, with KT88/6550 tubes. The poweramp input is a pentode feeding a split load phase inverter (sometimes called a concertina). Totally different from the run-of-the-mill Marshall or Fender poweramp design.
Calling it "solid state" sounding doesn't clarify the sound for me. Solid state distortion has a range of sounds, and not all of them are unpleasant.