You mean keeping your weight off the man's right arm, don't you? The man's left hand holds your right hand, his right arm goes behind your back under your shoulderblade and your left arm rest on it. I'm not sure I can really tell you how to support your own weight better. I think TheDitz has a good point about using your lat muscles to hold up your arms. My teachers have said a number of times that you need to make use of the lat muscles in keeping your dance frame. Her other remarks sound pretty good as well, it's just that I remember being told to use my lat muscles, too, so I wanted to second that one.
As for popping up on the 3, I'm going to assume you're talking about something like waltz where you end in an up position on 3. If you're rising too fast, hence popping up, you might want to make sure that you begin your rise earlier. A basic waltz Natural turn has rise and fall as follows (according to the ISTD technique book): commence to rise end of 1, continue to rise on 2 and 3, lower end of 3; commence to rise on end of 4 NFR (no foot rise), continue to rise on 5-6, lower end of 6. That's the man's part. For the lady's part swap 1-3 with 4-6 since you do the back half of the turn while he does the forward half of it and vice versa. If you're trying to stay down on 1 and 2 then you'll naturally end up popping up on 3 because you've got to do all of your rise in one count to get to the up position. Instead make sure you're taking all the time you have, two full counts or somewhat more, to do your rise. That way the rise will be smoother and you won't end up popping up at the end of it. If this isn't the problem or doesn't help fix it, maybe you can try to explain in a little more detail what the problem is so we have a better idea of how to offer help.
TheDitz, yes, learning how to use your weight kind of changes as you progress levels. I know I'm not the best dancer so I probably don't do everything I should, but often I try to work off what the lady's giving me. I can definitely tell when the lady's using her weight to help drive our motion. As you'd expect, it's the more advanced ladies that do this most often and it really feels different to dance with them. I'm not always used to helping to control that extra power/drive from the lady, but it's good to at least feel it from some of the women I dance with.

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James Marshall
marshall@astro.umd.edu http://www.astro.umd.edu/~marshall