Can anyone please let me know what universities have a LLM session starting in January?

There was this thread a few years ago:

https://llm-guide.com/board/europe/llms-commencing-in-january-february-2018-198795

However that focuses primarily on UK / Europe schools and since you posted in the USA forum maybe that's not your interest. For US schools, maybe check out GW Law, San Diego, and I believe American University WCL also offers a spring intake.

Go through this listing for more, but check out the schools listed there because some of the 'full-time' options may only start in the fall:

https://llm-guide.com/search/result?keyword=&area=15&language=1&program=1&intake=13/1