Hi Steve. Very common problem and the fact that you notice that it's going sharp is good. A lot of folks don't think much about tuning and it can ruin their sound. First, the 12 string problem. Very few capos are 100% effective in keeping the instrument in tune for the very reasons you outline but here's a couple of tricks I've worked out over the years. If you're willing to put a bit of effort into it it will pay off.

Use either a Shubb 12 string capo or a Dunlop C clamp. (which is my all time favourite because of the rubber they use)I think I've tried virtually every brand on the market and these two are the best. The Shubb is already slightly bent to fit a curved fingerboard but the Dunlop has to be put into a vice and gently banged with a hammer til it curves exactly like your fingerboard. (take the rubber off first!) Next rub a bit of chalk on your strings at the second fret, and put the capo on NO FURTHER THAN 1mm from the fret. Tighten it. Take it off, and the chalk marks on the capo will identify exactly where the thick bass strings are. Take a VERY sharp exacto knife and make a tiny triangular shaped cut to the rubber in the middle of the sixth and fifth pair bass strings. Don't worry if you screw it up the first time, the spare rubbers are cheap. When you put it back on ,the thick strings will sink into their grooves a little and hence not stretch as much when the capo gets to the thinner strings. This really works and I've modified dozens of capos for folks over the years.

The sharp E string is the result of either: You're fretting in the middle of the fret rather than very close to it..or your intonation is out.(check with harmonics at the twelfth fret) This can happen on expensive or cheap guitars. Sometimes the saddle has to be re-positioned or split. If you need more explanation on that, let me know. OOPS, I forgot one other possible (and often the major) cause. Your E string is coming off the nut too high and the groove needs to be deepened. Hope this has been some help.