Your story sounds quite familiar to me, because I was in a similar situation a couple of years ago. I really wanted to get research experience, but I'm pretty awkward, and was terrified of engaging my professors in any conversations, and even more terrified of asking them about research positions.
For me, the solution was taking courses that had a research component included. If you have this option at your university, I would definitely recommend it! In the second half of my third year, I took a research practicum course, and I was automatically placed in a lab. Although my first interactions with the PI were VERY awkward, I was already in the lab, so I had the opportunity to show that I was dedicated and hard-working, and eventually things became less awkward. The same thing happened when I had to choose an advisor for my honours thesis... I initially approached him because I ultimately had to find an advisor. At our first meeting I stuttered and probably made no sense at all, but because I was doing my thesis with him, I had the chance to show my abilities through my work.
If you have the opportunity, I would also recommend taking seminar classes. I was initially afraid to enrol in these, but they turned out to be my favourite courses. They also helped me overcome some of my nervousness around profs because there is a lot more student-faculty interaction.
Even if you don't have these opportunities, it seems like you're on the right track! Your interactions might start out awkwardly, but they will get better, and the more profs you interact with, the more comfortable you will become. I've noticed that some professors are not the most socially adept people themselves, and even if they are, they will probably understand your plight. In general, I think showing genuine enthusiasm for their research is more important than appearing calm and collected when you talk to them. Good Luck - You can do it!