Hi John!
You've already had a number of excellent suggestions, and I wouldn't argue with any of them. But I do want to elaborate on the issue of why you and your partner bump legs.
The key to not feeling like your legs are crowded, (assuming your dance position is correct relative to your partner,) is getting your center moving before your leg. This has already been said. What hasn't been said is that in order to do this your standing leg, (the one with bearing your weight,) must activate by compressing slightly and begining to move you, slightly, in the direction you wish to go. When your partner feels your center begin to move, she will be able to do the same, and lo and behold, she will move her leg at the same time as yours so that you don't bump her leg or feel that you have to push it out of the way. (This action is fairly subtle, and you will need your teacher's help to find the right degree.) This should also help you somewhat with the problem of leaning into your partner, since your base will get moving faster, but that will also require your teacher's assistance.
The annoying thing about body contact is that because you're afraid of stepping on your partner you become tentative, and because you're tentative you're more likely to step on your partner. Once you break this vicious cycle, body contact is the best connection possible.
I should add that there's nothing wrong with learning without body contact at first. Once you learn better how to move correctly, moving in body contact will become much easier. Have patience...many different concepts will help you with these problems. They can be challenging to learn, but as you do your dancing will become more and more enjoyable. 