One thing I like to do for queries like this is use weatherspark to look at historical averages. Here for example is a comparison of one spot on the way heading west (Las Cruces) vs. one spot on the way heading east (Mobile) -
https://weatherspark.com/compare/y/1...and-Las-Cruces[*]
My general expectation would be that heading west is colder and drier, particularly in some of the high desert areas. I also expect a higher day/night range so westbound you might get a below freezing cold but then warm up to 50-60F during the day. It can snow but my odds would put it slightly lower than 50/50.
In contrast, going east you have more chance of a cold wet day or two.
Over the last years, I've taken a few winter trips over the week between Christmas and New Years. For example, in 2019, I took Amtrak to Tuscon and then cycled from there to El Paso. Most of the days were fine but right at the end we had a cold front come through. It snowed just slightly with a strong westerly wind the last night before I reached El Paso -
https://www.crazyguyonabike.com/doc/22685 In 2018 and 2014, I went north instead and cycled from Brownsville TX up north to Austin. Nice temperatures most days but also a strong cold front with slightly freezing temperatures coming into Austin in 2014. I cycled "Christmas Ride" five times when they did it out of San Diego. Our route carried us up to ~6000ft - a little higher than the ~4500ft of Southern Tier but we had temperatures close to freezing most years and a snowfall one of the years.
I haven't done quite as much further east, but did cycle Florida between Fort Myers and Daytona Beach round trip on two winter breaks. Mostly very pleasant temperatures though one of the years we also have a cool 40F wind blowing. I was fortunate without as much moisture on those trips.
Overall, if you have some flexibility to stay put a day or two to let the worst of some weather pass - then either direction should be fine. I would expect heading east to be slightly warmer also with slightly higher probability of precipitation.
[*] Another weatherspark point westbound is Silver City, NM -
https://weatherspark.com/y/3086/Aver...tes-Year-Round This is near the continental divide and one of the higher spots along the route (assuming you avoid the pass and go via Deming). You can see average night temperatures just a little below freezing and a small amount of snow on average during the month of December.