Okay, my problem got fixed, and here's how:
My boyfriend also has an iPad (a 16 GB Wi-Fi, where mine is a 64 GB), and he recently restored it and updated it to 4.3.5 to fix some other issues he was having. He downloaded the same apps I was having issues with, and was able to proceed to the "Tap to enable tablet access." This is the point where I would normally get the Error -1004, but for him, it loaded the screen to enter in either the name/address or the account number. He entered my account number, and once there, it asked for the email address/password combination already enabled for Condé Nast magazine websites.
I went back to the apps on my iPad and, while I couldn't simply login (it would still show my already-bought issues as needing to be paid for again), I was able to tap on the "Tap to enable tablet access" button in the upper right, and this time, I didn't get an error. I entered my own account number, and then tapped to login with my existing Condé Nast email/password combo, and it worked! I'm able to see my current subscription and download issues.
I assume this will work for other subscription-based apps, as I've seen this error pop up for other non-Condé Nast magazines, like Time Inc.'s Fortune magazine, but I can't be sure.
You can only download one issue per magazine at a time; if you multitask away from the current download, it will automatically pause (and not restart until you press the resume button), but it shouldn't have to start from the beginning if you do pause.
For those of you without access to two iPads, I suggest restoring your iPad via iTunes. Make sure you've synced recently and have backed up everything that's important to you, or else have access to it via the cloud.