I replaced my 6 speed Suntour with a 7 speed Shimano freewheel. The Campy Nuvo group on my Raleigh worked will with the extra gear and my friction downtube shifters didn't care one bit. I think I had to play with some axle spacers to keep the chain from rubbing the stay near the hub.
The new freewheel only cost $18 back in 2011, so it's not like it was a budget buster.
Change your chain. They are inexpensive too in the grand scheme.