Even if each tier looks perfectly level, when I put each tier on it is no longer level but is tilting slightly. Not a great deal, but enough to make me nuts. What is the answer, dear ladies?

Everything you ever wanted to know to make, decorate and assemble tiered/stacked/layer cakes:

http://cakecentral.com/cake-decorating-ftopict-605188-.html

You might want to check out the illustrated common cake construction methods link (with complete directions) and especially the SPS sticky by leahs. (Or if using dowels, indydebi's illustrated guide to cutting them all the same length.)

HTH