Carol C. Yeah, that makes you a weirdo! ;DNo, seriously, it is a very hard medium to strike (between too much eye contact and not enough). There is lingering too long on any given individual, and then there is shutting down too far so that the audience (no matter what the size) feels that there is a wall erected between themselves and the performer.
What I have tried to do:
For large audiences, I mill my eyes about the crowd and make eye contact longer with those that seem to *want* it.
For smaller audiences, I many times will speak to them directly before the performance. Ask them their names, tell them a joke, try and make a connection to alleviate some of the tension that occurs when strangers eyes meet. And that really is what makes it uncomfortable. Try making eye contact with someone you know when they are on stage. It doesn't have nearly the impact that eye contact with someone you don't know at all has.
Another tactic to try: Sing with your eyes closed. It doesn't have to appear to the audience to be an avoidance if done right. Maddy Pryor used to sing barefoot, her hands crossed in front of her, eyes closed. It can look like you're trying to connect with your muse, if you put the right amount of feeling into the song. Of course, this works only for slower pieces. I can't see anyone singing with their eyes closed to a hard hitting "Pogues" tune, for instance. ;D
All in all, what you're feeling is completely natural. I have been singing in front of audiences for more than 20 years, and I still get the jitters, and feel awkward here and there. Your level of comfort does come across to your audience, so the more you can create "safe space" for yourself, the more your audience will be able to let their hair down and enjoy as well.