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Click here to see what I use to tow with my 1996 LT4 coup. The same hitch is supposed to fit the '89 too.
Price and quality are decent but installation will require drilling a few large (1/2") holes in the aluminum sub frame that supports the gas tank. I tow a mini utility trailer and have hauled a number of machine tools, tires, 60gal air compressor, and more.
Have a helper when fitting the hitch so you wont go crazy trying to keep the thing still when doing the hole position layout. It's not light weight. There are spacer sleeves and washers used to set the correct height of the receiver. Be prepared to 'fiddle' with it to get everything perfect.
Years ago Reese or Curtis sold a hitch, they were a little invasive because it called for drilling through the bumper/Gas Tank frame extension. So on that score it gets a leave sport cars sport cars. Maybe other hitches are out there that bolts directly to the hard frame without drilling but I would not go any larger than a Single Wave Runner pretty much any Corvette
Curt still makes a hitch for a C4: C11705. You can still buy this from places like eTrailer.com. The holes that need to be drilled are in the aluminum crossmember, so it won't be very difficult to drill them. It's a Class 1 hitch, and the limits are 200lb tongue weight and 2000lb total towing weight. Your 350lb camping trailer should be no problem at all. Lots of people tow small trailers with cars like this when they need to haul competition wheels/tires to out-of-town events. They would be at or slightly above that weight when loaded, and there are no problems with that.
Here's what I use...
