Not sure on a 1998 or if it's a sled, but on my 2005 400 there is a rubber diaphram on the primer. Mine was leaking so I bought a new diaphram and it solved the problem.

Gas wise, Is there any gas coming out of your carburetors overflow? Are you sure it's the primer leaking and not the overflow valve?

Sounds like your carb is running really rich. The excess fuel drips down the cylinder and gets in the oil. It's normal for some gas to be in the oil, but a quart is probably way above normal.

 

 

It's old, never having the gas off, and with gas having ethanol in it... When's the last time the carb was broken down and cleaned? Could be a lot of rubber in it deteriorated. I'd question the integrity of the float too.

If it were me, I'd take the whole carb off, break it down and inspect it all, then clean all the jets and give it a carb bath. Replace any seals/diaphrams that are shot. including the float if it's bad.