It might depend on how far along you are with the build. Less than 1/4 of the way, I might get a new board. The good news is that the trace is simply a wire stuck to a board, meaning that you can abandone it on both ends and run a wire from where it was supposed to go. Make sure that both ends are cut.
Just as importantly, it is possible that you are using too much heat with the soldering. How many watts does your solder iron have? Is it adjustable? What tip are you using? These are important questions trust me. I did some horrible work before I bought a good soldering station. Since you are removing pads and traces are coming loose that is important to consider.
 I won't say that the board is defective, but it is possible. When you were getting the pot out, what did you use? A solder sucker? Solder wick? Sorry for all the questions. Yup been there done that. It all matters.

Sorry for the late reply. The information that you have supplied tells me a few things:

1) 775 F of heat is way too hot unless you can get in get out fast. I run my iron at 650 and that is considered pretty hot too. The melting point of a lot of solders is between 380-450 degrees, so try experimenting with a much lower temp to see what happens.

2) The right tip will ensure the best heat transfer. I typically use a 2-3 mm chisel tip so the heat transfers more evenly. It works for most applications.

3) Solder wick is your friend when extracting solder from multiple connections. The solder must be removed from each connection as in the case of a volume control. Then bit by bit it may wiggle out as you apply more heat to the pins.
Fwiw, there is a product that is called Chip quik. It is actually an alloy made mostly of Cadmium. The melting point is low at about 200 F. So then you apply it to the original solder on the pads and it mixes with it. Now you can melt the 'new ' bunch of solder at a much lower temp, and it will stay molten much longer, giving you extra working time and ease with the solder wick to pick it up.

3) It's not likely that the traces can be repaired easily, but don't worry. A jumper wire can run straight across the circuit board where it ends up at the far connection. Make the wire long enough so it doesn't get in the way of anything or pick noise from nearby circuits.

You will be fine.

UPDATE:  I tried running a jumper from the POT to each RCA input and also to each 1K resistor to the input pins of the tube.  I lifted the pads on every interface that I jumpered and pulled the trace of a bit and snipped it. (see pic)

Couldn't get it to work.  One channel was non-responsive and the other was always on, regardless of the position of the pot.  

It was worth a try - had nothing to lose.  

thank you again for your advice and encouragement.  Will redo once the new boards come in.

Interestingly, I was thinking the same thing about point-to-point!  That will be my next amp.  Thanks again.  Cheers