Ok first of all it will depend dramatically on what year model as I have never seen a starter relay on any of the models here that look like the one pictured.
If it is a 94-01 then the NSS WILL ABSOLUTELY affect the overdrive as it sends a signal to the TCC to say what range the transmission is in, this style of NSS is very easy to dis-assemble, clean, re-lubricate and re-install.
Gojeep's site has a good write up on how to perform this task....