Is it wise to quit a job if you don't even have an offer letter?

I’m very new to this so sorry if I’m asking a bunch of work related questions. They expect me to start January 6th if all goes well with my background, drug, and TB test (which it should). Results will be next week.

I would have started earlier if it weren’t for all the holidays coming up. But anyways, should I tell my internship NOW? This gives me about 3 weeks until starting. I don’t have an OFFER LETTER though or any contracts with my signature for the new organization I’m going to be with =\