It is the sealing rings on the servo. Positive about that. With only 4K miles on it and sitting that long the rubber seals on the servo have hardened up. If it was the clutches it would still shift but slip all the way into gear. Letting off on the gas shows it is the servo
The servo works like this. It applies 2nd gear on the 1-2 shift. But the kicker and what many people don't realize is that it must *release* 2nd gear before 3rd can apply.
The piston (in the high gear drum) is trying to apply 3rd gear. The band is still holding onto 2nd so the high gear clutches slip. The valve body diverts fluid to the back side of the servo to make it release. The servo still has pressure on it from the front side. But the seals are leaking and it doesn't move fast enough. Or as fast as the high gear piston applies the high clutches.
When you let off the gas it reduces the fluid pressure to the servo. Under less pressure the fluid isn't blowing by the seal as bad, the servo releases the band and 2nd gear and it shifts to third.
Fairly simple to fix. You can do it with the trans in the car. Loosen the locknut on the adjuster, tighten the band all the way down. If it's loose the adjuster mechanism may fall off inside the trans when you remove the servo from the other side. Unbolt the servo cover, remove the servo. If your trans is original it will have rubber o-ring type seals on an aluminum piston. If it is a later model trans it may have an "n" servo. Looks like a hockey puck on a stick. No o-rings, just the entire piston is covered in rubber. In that case replace it ~$10. Replace servo, re-adjust band.
Been through this on 2 C-6's of my own and one in a friend's car.