Can any one recommend a good app or procedure? I've googled this, and searched manually, but so far, no dice.

Ideally, I'd like to do this wirelessly, and without SD cards. Failing that, using a PC as a mid-point would work too. I've tried the suggested apps I've seen, plus others, here's the ones I've tried so far:
SMS Backup from George Android
Advanced SMS Backup from beginnow
SMS Backup+ from Jan Berkel

I thought there were more, but they must be hiding in my "all" list under the Play Store.

The last one in the list won't do picture messages. I think the first one sent an email to gmail, but wasn't able to do anything with it from the other side. And the other one claimed to back up everything, inluding pic messages, but then could not restore them on the new device, kept giving errors. I'm afraid I didn't document this all very well.

I've been trying to go with free options, because I don't want to buy something that won't work, however, if something is known to work with these 2 devices, and does pic txts, I'd be willing to spring for it.