It's too early to know for sure, but I think I may have just found a fix! (I really hope I don't jinx myself...)
When I deactivated my old Droid and activated the DNA, all the WiFi connections that I had ever used on my old Droid were automatically transferred into the DNA, so when I was looking at the list of WiFi connections there about 30 or so I could choose from, ranging from old subway WiFis to restaurants to my friends' houses. Last night I went through and manually "forgot" each of the ones that I had used on the old phone, leaving basically my home and work WiFis as the only ones appearing as options. After doing so I was able to connect to my home WiFi just fine, and I didn't lose the connection for hours (before applying this "fix", I would lose it every few minutes or so). I kept my WiFi on last night while I was sleeping, and it appears that I didn't lose the connection once.
Again, it's too early to tell if this really fixed the problem for me; I'm interested to see what happens when I try the WiFi tonight when I get home from work, but I am taking it as a positive sign.
Let me know if this works for anyone else!