Podcasts always "Unable to download at this time", despite retry, reboots

I have an iPhone 4 (not S), running 7.1.2, which is the latest version of iOS that will run on it. I know that means that eventually it's going to stop working, but I can't afford to shell out on a new piece of hardware anytime soon, so I've been trying to baby it along.

Starting with the new Podcasts app, I began to get occasional "Unable to download at this time" messages, but clicking again would download them, or failing that, turning the phone completely off and then on. Once I went a whole week without being able to download, but it turned out the phone was full -- once I downloaded some excess photos and listened to some hour-long episodes, they downloaded just fine again.

Starting this week, nothing will download, no matter what I do. I have a full gig free. I have rebooted it several times a day all week,

I do not use iTunes to sync my downloads, because the desktop version refuses to mark episodes played that I listened to on the device, so the instant I start iTunes it starts downloading hundreds of episodes that I've already finished long ago. Because of this, I download episodes directly to the iPhone over wifi as my primary method.