If your balances are absolutely at zero, i.e. no balance on any card, you're going to have a noticeable score ding once all the zero balances hit your credit report.
With your scores approaching 800, losing 15-20 points may not be a big deal. Otherwise, to avoid the ding, let one of your cards report a small balance from a new charge. Do this on a card where you weren't paying interest.
Don't get a mortgage without any purpose. If you don't have any installment loans at all and want a loan for the sake of a score bump, the share secure loan from Alliant described in the scoring forum is the way to go. I wouldn't think you're in any need of a score bump to get nice cards, though.
If you could tell us which cards you already have, their ages, and their limits, people here will be better positioned to give you advice on which cards to seek out moving forward.
