I did tons of this type of gig in my early years. Started, literally, playing on the streets. Got invited to some open mics. Ended up running open mics then booking actual shows with other solo artists and so on.
It never really got easier... especially with original material. Total nervebait, pants pooping terror that you can't show or let affect you during the set.
My recommendation... play some covers to fill out your set. I always found I was far less tense when playing other people's material and it made the original stuff easier because it wasn't non stop naked ass soul baring for the whole time I was on stage.
The most nerve rattling solo gig I ever played was a funeral. My friend's dad had died semi unexpectedly while in hospital. She had been to quite a few of my solo gigs and apparently the family had been meaning to ask me to drop by the hospital with my guit while he was alive so we could just jam out. He was apparently an old timer guit player into 50's/60's rock and country which I actually did a bit of for my solo act. My Johnny Cash covers were of particular interest. Unfortunately he just passed away suddenly before they got around to asking me.
So my friend asked me to come and play at the memorial service for her mom (in particular because he used to play for her) and the rest of the family. Wow. Complete out of the blue honor and absolutetly terrifying prospect... but as much as my nerves were shreiking "NO!!! OH HELL NO!!" how could I refuse such a request.
They gave me a list of specific songs that he liked and would play for for his wife (some Elvis and Buddy Holly... that type of thing) with a specific request to learn "Love Me Tender" (that was the ultra special song he used play for the old girl). I had a little under a week to learn the new material and rekerjigger my usual performances of other tunes I already knew to be less bombastic than how I'd normally play them.
When the day came I was all freaking jitters. I shined my boots, combed back my hair (ah Johnny, we miss ya) and put on the only semi respectful clothes I owned (all uncomfortable and ill fitting dress wear accumulated over the years from pawn shops and X-Mas presents from my mumsie). I even wore a freaking tie.
This was also during the first era of my crippliness (my back was an absolute disaster area and my left leg was a quivering mass spazm-y pain). I packed up my guit, hobbled my ass in the city heat multiple blocks to the sweaty half hour streetcar ride to a semi skeevy neighborhood I'd never been to (where the funeral home was). By the time I got there I was not quite as fresh and dry as I was when I left home. I walk in and am greeted by my friend and then some of the family. Being the gutter rat I was at the time I was extra self conscious about coming across as some kind of lunatic to these rather normal folk. So I was smiling and shaking hands and just trying to keep from tripping over my own dick.
I asked where they would like me to set up and they led me into a big room and pointed to a chair and music stand they had set up for me. Again, I was pretty nerved up so I just hustled myself over, sat down, unpacked my guit and lyrics. The family started gathering around (there was under a dozen close family members there... they wanted to keep it small which kind of made it even more nerve wracking). The wife/mom sat on a couch that was directly to my right (she was essentially sitting right next to me) and some brothers/uncles while everyone else pulled up chairs creating kind of a circle. No one was more than 4-5 feet away from me so VERY intimate crowd (good thing a toned down the songs otherwise I would have been literally screaming in people's faces).
It was at this moment that I looked behind me (in my nerved out stupor I totally missed it as I walked up) that there was a coffin RIGHT behind me. Like I could reach over and touch it. It was closed but at that moment my nerves went into overdrive. Total internal freakout. "MUST... NOT... F*CK... UP!!!"
Anyway, they were all extraordinarily nice people. It was ultra intimate. I'd play a song then we'd talk. They told me about him and did their own remembering. No tears or wailing. Just good old Canucks remembering their dude. The songs came out okay and seemed to lighten everyone's spirits. Totally casual and friendly.
Then came the biggie. "Love Me Tender" for the mom. She was such a sweet lady and a total trooper. You could tell she was sad but completely cool and pragmatic about it (like they had a good life but all things shall pass type of vibe).
This particular tune was absolutely new to me. Of course I'd heard it a million times but had never played it until that week. It's a really delicate song already but within this context... well, some random street punk playing it directly to a greiving widow looking straight into your eyes... yeah... crazy stuff. ULTRA NERVOUT at that point.
Everything was going fine up until I think the second verse when my brain just said "SCREW YOU! I'm outta here!" and forgot the words. Well... A word. It was probably only a 2 second stumble but the mom/wife, who was hanging off every note/word, with some soul penetrating kindness prompted me by singing the words I was fracking up until a got back on track and finished the song.
I was mortified but of course none of them cared and were just so kind and appreciative. Very nice folks.
As I was leaving my friend came up and handed me an envelope with $100 in it from her mom which is what they said they would pay me for the gig. It really wasn't about money (I did of course need it but that's not why I was there). I tried to refuse it making the excuse that I screwed up the most important song (I just didn't want to take the mother's money) but my friend and then the mom insisted. That it was a pride thing and even if I was just there to play for a friend and her family I had to take the money so I didn't push the issue, thanked them profusely and went on my way.
I guess the moral of that story is... you just gotta plow through it and choke back the nerves as much as possible.
Incessant practicing of the set of course helps immensely. Like to the point that even if you crap yourself on stage from nervousness you know the songs well enough to just play right the frack through the stink...
Then blame the smell on skeeviest looking person in the joint (this ploy doesn't work as well if you are wearing light colored pantaloons).
/has never crapped himself on stage... but has come close... lulz