Unfortunately, I do care how I look while I'm dancing, because even though I want to connect to the music, I'm not going to sit here and lie and say I don't want to connect with my partner, too. And the way I look definitely influences the quality my dances, and the subsequent high I, and my partner, get. It's not limited to how I move and dance, but also how I dress.
You lost me. It's how you
feel that affects the connection with your partner. How you look certainly reflects the quality of your movement (and timing and other things of course), but form your follow's perspective how you look is just a proxy for the real thing -- how you
feel. Every follow knows guys who look amazing on the dance floor but feel like crap to dance with. And they're not exceptions either, they're more plentiful than you'd think.
Here are two videos of me dancing with two New York guys who are two of my favorite leads in the world and have been so for several years. These guys feel so amazing to dance with that I would choose a dance with them over a dance with any of the famous pros any day. Do they look good? Sure, they do, but if you put them on an international dance floor with other good dancers they would probably be lost in the crowd. But the way they feel? The way their movement and lead are
so amazingly in sync with the music? The smoothness of their lead and the tension build-up and release in their movement? The way they playfully express the music? It's a heavenly feeling to dance with them!
So do use how you look as a way to see what you need to work on in the way you move (especially your basic step), but remember it's a proxy for just one thing--the quality of your movement. It's the combination of your lead quality, movement quality, and--above all--connection with the music (both in your movement and your leading) that will make or break the connection with your partner.
So, guys, the musical connection you mention you want is not an "even though", a side thing with regard to your partner connection--it's a necessity if you want to feel good to your partners. From my perspective, it is impossible for a lead to feel good to dance with if his connection with the music is off.
Here's a specific musical leading tip for you guys that for me takes the dancing to a whole new level and that is fairly easy to learn to do: look at how the guy in the second video is giving me a double spin at 1:34--notice how he is not just leading the double randomly (like 90%+ of guys usually do), but is leading each turn to the music? That's what I'm talking about when I talk about the leading being in sync with the music -- and it feels amazing to dance with a guy who does that
I found that paying attention to how I looked took away from me paying attention to how the music made me feel (if that makes sense).
Has anyone else felt this way about their dancing?
Is there a way I can do both?
We
all feel like that when we see our videos.

Especially those of dances we really enjoyed. A dance or partner may feel amazing to you (like the guys above are to me) and yet the videos may look just ok or even mediocre and totally not capture the feelings you and your partner had during the dance. So don't worry about it, use the videos as a way to improve your basic and any other issues you observe, but don't think how you look in a video is the be-all-end-all, keep enjoying how you feel when you dance. It's all that matters.
And yes there is a way to do both, in fact the feelings and the quality of your dancing usually go hand in hand: as you improve your dancing and your partner connection (lead), how you feel will also improve. Thinking back to my first couple of years dancing, I thought I was having fun but it was nothing compared to the feelings I get now, because I am much more relaxed and better able to express the music and connect to my partners (and not have to worry about things like keeping my balance in a double spin

)