Yes @Hugh Jars, I have been self-conscious and anxious several times at a driving range and even on the golf course. It was worse when I was just starting out but it still lingers with me today.
Unfortunately, I have only found one way to handle it:
The best way to get over it, is to simply GET OVER IT. Meaning, you have to keep going. Eventually what is uncomfortable can become comfortable.
Do not shy away from the range. You need the practice. I have to keep reminding myself that the people around me don't care!
Play with more and different golf partners. Play with better players, if possible. Go out with a group of buddies who might tease you a bit. All of these things probably could trigger some more anxiety. So long as it is bearable, just forge on. It's all in good fun.
If all else fails, you can get a decent net and turf mat for less than $100 to hit off of in the privacy of your own back yard but that creates many challenges to effective practice.