Boy, that's a tricky one. Obviously it's complaining, probably accurately, that the battery doesn't have enough charge to work. I think you've done a good job covering the first step, that of making sure your power source is good by using a known good cable and charger.
That leaves us with the two parts you're suspecting, the charging port and the battery. To be honest due to the reaction of the port to your wiggling, it sounds like the reason the battery isn't getting charged is that you haven't got a good connection between the charging port and the plug on the cable.
However, I'm going to suggest trying a new battery first for the simple reason that it's easy to replace the battery and really really hard to replace the charging port. I'd want to 100% eliminate the possibility that the problem is with the battery before I even considered trying to replace that charging port.
There are a couple of issues with replacing that port. First, of course, is the difficulty in soldering a 30 pin connector. That's a lot of pins to get soldered on correctly, so there's a lot of room to make a mistake unless you're fairly experienced at soldering. The second part of that issue is that of locating a replacement port; I did a somewhat cursory search and came up with bupkis, nada, nothing.
So yeah, I'd say your next step should be to swap out that battery. If that doesn't work there's a small possibility a different charging cord may work better; you could try and locate one that's being sold specifically for your Sansa as a suggestion.
Try those out then come on back and let us know what you found. If it's still not working we'll see if we can figure out where to get a replacement charging port for your device and how to replace it.