Last year I stopped paying a monthly premium for one of my kids. They had switched insurance coverage, and I had been paying the policy manually, and we were busy, so I just didn't worry about officially cancelling the policy for a few months.
Eventually we received a letter from the insurance company saying that since we hadn't paid the policy for those last three months, if we ever wanted to re-enroll in coverage with them again, we'd have to pay those premiums first.
They did offer an option for us to write them a letter and request them to change the cancellation date. Because of the "burning bridges" aspect of things, I did go through the hassle of writing the letter and sending it in and making sure that they did backdate the cancellation date to the end of the last month for which we had paid.
In our case, we included in the letter the fact that we didn't make any claims on the policy during that limbo period, and explained that the kid was a busy college kid who had been quarantined and sick with COVID for about a month during that time, and in general just tried to be apologetic and professional.
I'm sure the insurance company just doesn't want to allow the moral hazard of people not paying their premiums for a while, waiting to see if they actually needed the coverage or not, then paying the premium or not as circumstances warrant. I think your case is similar to mine and you'd probably succeed if you write them a letter and request that they terminate your policy at the end of January.
I've never seen a health insurance company report on a credit history - they're not extending you credit, so I'm not sure why they would. But in theory you could check yours for black marks - it may take some time for the black mark to show up so you might check in a few months just to be sure if that's a serious concern.