Dancing should be a place where everyone can feel comfortable and have fun. You should never feel pressured to do something that is outside of your boundaries. Ever.
I'm glad you brought this up because this is a very important issue that often either isn't talked about or is met with a lot of judgement.
My husband and I have a similar situation. He feels the same way as you. He enjoys it, but from the beginning his goal was just to dance with me. He never wanted to dance with multiple partners and has always enjoyed sitting on the sidelines watching and only dancing occasionally with me, but letting me dance with others - and even enjoying seeing me dance. I totally get that, and respect it, and just appreciate so much that he makes the effort to dance with me because he knows that I love it.
So... the problem isn't with your desire to sit on the sidelines and just dance with your partner. My guess is that you just need to be able to express your feelings more assertively to the ladies and your partner. What everyone else said is completely right - they're asking you to be nice and because they love to dance and they can't imagine why someone would just want to sit on the sidelines and watch when you could join the dance! If you are sheepish about saying no, then they assume that you are just being shy or are nervous about your level but really want to dance. So, they apply the golden rule - what would they want someone to do for them in that situation? They'd want to have a little push to help them get out of their comfort zone. I'm not saying that you need to be an $%&*@! about it, just say with a smile, "No thanks. I'm enjoying just watching." Or something. Repeat with confidence as many times as necessary. You said you've always been shy (me too) - a book that has really helped me with this is one called "When I Say No, I Feel Guilty." I HIGHLY recommend it.
My husband is very assertive, so it generally is never a problem when we're somewhere. The girls are never hurt when he turns them down, and they know better than to push it - and frankly, so do I! There was only one time when we went to a class before a Tango night on a Friday night when the instructor made it uncomfortable for us to stay together. He was pissed because it was supposed to be a "date", but we ended up not spending very much time together at all. The solution? We never went back. We found a different venue to dance at. The biggest problem that we have now is that he's so formidable that he has to walk out of the room before anyone else feels brave enough to ask me to dance! Lol
Getting the other girls to stop pushing you is the easy part - getting to the point where you and your partner can come to an understanding is the hard part. To do this, you're going to have to have some deep heart to heart conversations. You need to find out why it bothers her that you like to sit out. You've got to listen without judgement and take deep breaths to keep from getting frustrated so terse words don't accidentally sneak out. Listen and seek to understand her first before you try to get her to understand you. Ask probing questions. Don't get defensive, and understand that she probably will be defensive if this is something that has been a hot topic before. She may feel guilty because she assumes that you're not having fun when you're sitting out. She may feel threatened / guilty because she's dancing with other men and is afraid that deep down you're getting jealous. She may be disappointed because she imagined that you would be the next Fred Astaire and she would be your Ginger Rodgers. Maybe she feels confined because she feels that her level can't progress unless yours does.
A quick side note on that last one - it's a complete falsehood!!! My husband and I have excepted that it would be impossible for him to ever reach the same level as me, and that's okay. I've been dancing my entire life - I even used to teach dance. There's NO WAY he could ever bridge that gap. But, we can still have fun dancing together, AND I can still progress in my dancing. The way we've worked it out is that we take a lesson every other week together and I take one just by myself on the weeks we aren't doing one together.
The worst thing you can do is grin and bear it until your discomfort eases! Your discomfort WILL ease, but only if you face it. Both YOU and your relationship is going to be stronger for it.
Hope this was helpful. Good luck!!!