WD-40 is your friend. Your volume knob acts like a variable resistor or rheostat and is referred to as a wiper. It sounds like it's just dirty. Open the unit up and be careful as you use some compressed air to clean out the chassis as best you can.
Short and soft blasts should be enough. Then, after you let it air dry a bit (some minor condensation will occur), direct the WD-40 with the straw as best you can at the backside of the volume knob and gently spray a small amount, and then give your knob a few turns back and forth.
If you can, give it a listen before closing it back up as it may take another dose or two. In the neighborhood I grew up in, there was a neighbor who repaired TV sets for a living and his son told me that half the time, all that was needed was to take off the knobs and clean them out with WD-40 and his father would charge them for something else.
As for whether a newer integrated would sound better, I'd say yes. 30 years is a long time for things to age.
All the best,
Nonoise