May I please ask a difficult question, did you buy the part from a Yamaha dealer or seller, or from ebay or such. Thank you.
When I questioned your starter solenoid post and you clarified starter relay, and you kindly posted the picture, I saw the battery connection is on the correct side, and I attributed the rubber boot damage to being understandable on a nearly 20 year old bike.
If you bought this starter relay from a Yamaha seller or dealer and specified Road Star Warrior, then I would not suspect what follows: if you bought it elsewhere then the relay circuits may not be correct for RSW. These relays look alike in many cases, but subtle differences exist and many sellers make assumptions because they are clueless.
If the circuits are wrong then you might have somewhat melted a wire somewhere along its length (even the wires coming off the plug).
Also, if the starter continued to run while the motor was running then potential exists for Sprague clutch (anti-reversing gearing) damage, starter gear damage, and other general starter damage. If any occurred, it could take time to become observable.
You might do a motor oil and filter change and carefully inspect all the oil, including the oil poured out of the filter, for steel or aluminum shavings.
Also, while riding, just pay attention for any increased lifter noise from metal plugging the hydraulic lifters and push rods. And of course if the clutch starts slipping due to embedded metal shavings.
Probably you're fine.
Hope you let me know where you got the (faulty) (wrong) starter relay. Appreciate it.