Madrid + One City in January - Sevilla vs Valencia?

This is a pretty open-ended question.

What I read as your nominal criteria has less to do with tourism and more to do with livability.
Not sure that many folks on the forum have that level/duration of experience to comment. I would
not necessarily pick the best tourist city as a place to live for a few months.

Plus, if you head to Sevilla, you have all of Andalucia at your feet and you're only spending
2 nights. For that criteria alone, I'd say Valencia.

Which is the better city overall?

That’s a personal preference question only you can answer. To save your limited time I’d go to the one that’s easiest and quickest to get to. Visiting a place looking at it as a place to live is a lot different than looking at a place to visit for a few days as a tourist.

Toledo and Segovia you should do from Madrid on day trips.
Seville over Valencia

Truly, I think both cities would suit, and both are lovely in January.
For me, Sevilla has a slightly more small town feel than does Valencia, but that's highly subjective.
For walkability I'd give a slight edge to Sevilla.

I've lived in Valencia (only for two months, though) and been to Sevilla numerous times, most recently last year, for about a week.

I agree with EKScrunchy. Both are lovely. Slight edge to Seville. In fact, I think EKScrunchy told me about key attractions in z Valencia when I was planning that trip. Could you cut Madrid short and do both?

Good idea! Since you are scouting for a place to possibly live in the future, you could consider cutting the time in Madrid (usually a little cold in January) and spend time in both cities. I'm not sure I'd go l the way to either city just for two days. When I first read your question, I missed the fact that this would be only a two-day stay. Can you really get a good idea of whether or not you might want to live there if you only have two days to check it out?

I do know a couple from NY who have been spending winter in Seville now, for a few years...they also looked around (in the south) and settled on Seville.

If you do decide on Seville, you might want to take a food tour with Shawn Hennessey; she is from Canada but has been living in Seville for about 20 years now. I bet she could really fill you in on what it's like to live there...or to spend a few months a year. I took a sherry tour with Shawn last winter and it was lots of fun, with great food and wine....I learned a lot.

https://azahar-sevilla.com/sevilletapas/tapas-tours/tapas-tour/

We visited Madrid, Sevilla and Valencia in January/February.
https://community.ricksteves.com/travel-forum/trip-reports/spain-in-january-february-madrid-sevilla-cordoba-toledo-valencia-and-coast-n-s-of-alicante
If you just wanted a change from Madrid for a couple of nights I'd go to Toledo.
For just two nights to get a taster for a longer stay I'd be swayed by what time of year your longer stay is - Seville in July and August wouldn't be a good idea. If your longer stay will be in winter I'd choose Sevilla because there is just so much to see there and nearby, and all of it is wonderful.
Valencia is a very pleasant and liveable city but hasn't got as much 'wow' factor as cities in Andalucia.

featured in the current issue of Smithsonian Magazine -

The Oldest Restaurant in the World Just Turned 300 Years Old
https://www.smithsonianmag.com/travel/the-oldest-restaurant-in-the-world-just-turned-300-years-old-180987899/

Madrid’s Sobrino de Botín is filled with legends, ghosts and priceless art. The food is pretty good, too.
Legend has it that 18th-century Romantic painter Francisco Goya was once a porter here. Ernest Hemingway set the closing scene of The Sun Also Rises at a table in an upstairs dining room, and the signatures of Spanish kings throughout the centuries adorn one of the walls. There is also most definitely a ghost in the wine cellar.
Sobrino de Botín, confirmed by the Guiness Book of World Records as the oldest restaurant in the world, just celebrated 300 years of scintillating history.

In general I say Sevilla because it’s a bit more amazing scenery and sights in a dense area.

For trips with kids I say Valencia for the awesome zoo and aquarium.

Overall, you can’t go wrong. Both are great.