The nervousness drastically reduces with time - at least it has for me. When I started at open mics & sessions I'd sit there terrified of my turn approaching - which meant I missed out on what everyone else was doing. Now it bothers me not one bit - though sometimes I get nervous if I'm going to do something new or extra important - or if I'm talking to someone else who is nervous themselves...
I try to have one "throwaway" song to do first - something that is in exactly the right pitch for me & is easy to sing etc. So then I don't need to be nervous at all about that one. Once I've done that one, I'm in performerly mode & so don't get stressed about the more difficult ones.
I very rarely drink if I am going to sing, as I like to have total control over what I am doing, and I like to change elements of the song as I go along & I couldn't do that successfully after a drink (I'd wander off and never get back to the right tune/timing). The fact that I am sober helps me to NOT be nervous, but dint of the fact that I know I am in control. The more confident I am the more in control I am - so you can end up with the opposite of a vicious circle (what would that be called?).
Sometimes, though, for no good reason, I'll have a night where I just know nothing is going to work right, and I feel nervous, and perform poorly. I try to appreciate that these are isolated glitches, not a trend. So don't assume that because you are nervous one night you will inevitably feel the same the next time. Similary - just because you have a fantastic time one night, don't assume that everything thereafter has to match or exceed it.
PastyJane - unacompanied singing is a thing all of its own - you don't need to be accompanied if you don't want to be. Do it with conviction and it can be far more effective than the same song with guitar etc. It makes a nice break from an evening of entirely guitar+voice songs. You can do a lot more with a song vocally if you are not having to fit in with an accomaniment - go for it and enjoy....
There's tons of great advice in this thread - especially from Sophoclese and Hesperis. There must be old threads on this subject too, if you poke about a bit.
Best of luck BobS - I'm sure you'll find your fear will drop off exponentially with the number of times you perform.
I'm not totally convinced about the need for any stage-fright/nervousness at all - though I know it's often said that there always has to be some. I think its probably different for different people. I'd hate for anyone to start worrying that if they aren't afraid then there is something wrong......

Kris