If you have a multi meter you have to take the off the wires on the pot, after marking the wires carefully so you know which lug they re-attach to. Don't forget to mark them.
Then turn the pot all the way to the right(clockwise) and with your meter take an OHM reading on both outside lugs of the pot. That will give you the pot's rating total ohm value.
If you want to see if the taper is still good then connect one lead to any outside lug and one to the middle lug and turn the shaft slowly from one end to the other, watch the meter to see of the numbers change smoothly without jumping or hitting dead spots where there's no reading at all. You can do this reading from the back without removing the pot, but don't hold the pot or touch the probes to your hand or the casing while you do it because that will give you a false reading. The tone pot's value in the original ts-808 and ts-9 should be 20K. I think the reissues use those values as well. The pot should read very close to 20K but if it's drifted a bit it might read as low as 18K which still wouldn't be a problem. The wipers inside those pots which controls the resistance as the shaft is turned is 1/32" or even less so it can wear and drift over time.
As far as actual sound, pots can affect tone somewhat. If you buy another I'd replace it with an "Alpha" pot. They're sturdy, have a smooth taper and will keep your pedal's tone perfectly intact. They're very easy to find as well. They also make pots for other companies so it's not hard to pick one up. As far as the pot value, yes, an Alpa pot has a stamp on it, either on the back where the alpha stamp is or when you look at the face of the pot(shaft side) you'll see a little white stamp on the brown plastic that says 20K at the top of the pot indicating it's rated value.
One more thing, when cleaning your pot use deoxit, not wd-40 or some other kind of oil based crap. The wipers in these pots, unless custom made like the RS carbon super pots, are silkscreened and can be dissolved or gummed up with flat spots if using the wrong type of contact cleaner.
wd-40 is great for door hinges and squeaky car parts but don't use it on electrical components. I've had to re-clean number of pedals and amps when the owners sprayed that stuff in because a friend told them it works.
When spraying the deoxit into the opening on the back(side) of the pot where the lugs are just spray a tiny bit in quickly and then turn the pot back and forth a number of times. That should clean and lubricate it properly. Too much and you'll have small lake of cleaner in there that will take way too long to dry and will attract more dirt.
Your pedal has diodes and transistors that can be sensitive to shock and from what you describe it sounds a bit like a faulty diode but it could be the pot.
Good luck with it.
Sorry for the long post but I wanted to make it as easy to follow as possible.