@arbaman: Thanks for your response. Yes, I'm having a crash course in batteries and their management. I'd heard that it is a good idea to cycle the battery now and then, but didn't understand how crucial it is to do so.
I spent a couple hours on the phone yesterday with Apple, talked to five different people before a Senior Advisor advised me that Apple simply does not repair 2014 Mac models any more, as they're now on the "Vintage" list. Oh. So my experience with the 2015 MBPro last summer cannot be repeated with this 2014 MBPro.
So I looked around locally, found a guy who looks good – and who says he has the battery in stock, and can do the job for $200, less than Apple would have cost, and a lot less trouble. But when I took the computer to him, he suggested trying several forced complete cycles on the battery, saying he'd seen cases where that solved the swelling problem. A good sign, I felt, that he didn't just take my money, but tried to help me first.
So I'm trying that, though I suspect I'm going to have to go ahead and get the battery replaced. So far I have determined that when running the computer in Target Disk Mode (his suggestion), the battery lasts about 2½ hours, which is certainly not good, though it wouldn't matter much since I never use it as a portable. He even said he could just disconnect the battery, or take it out without replacing it, if I'm using it always connected to power. But maybe I'll need to take it somewhere someday.
As I wrote, I also have a 2013 15" MBPro – which is what I'm using now while dealing with the 2014's problems – whose battery is very swollen. I'm thinking I may try doing the replacement myself with iFixIt's kit (it also needs RAM replaced on the logic board) one of these days. I've been struggling with chronic illness for some years, but improving lately, so hope to have the energy to take on such a job at some point – so long as it's not my primary computer.
I also have a 13" 2014 MBPro, whose battery shows 756 cycles (which would be one every 3-4 days throughout its life), but looks like new (I haven't determined how long the battery lasts, but will soon). Anyway, I'm resolved to start cycling batteries on a regular basis.
Thanks again.