You asked a question which hasn't been directly addressed in other answers yet:
And get the benefits?
Many employers run benefits on monthly or bimonthly cycles, so if they record your last day as December 31st, you may indeed lose your benefits as of that date.
Your new employer may also run benefits monthly, in one of two ways: either your benefits start as of the 1st of the month in which you begin work, or your benefits start on the 1st of the month after your start date. Given that you've stated you begin your new job on Jan 6th, this may be either Jan 1 or Feb 1, respectively.
So, the first step should be to reach out to your new employer and confirm their benefits policy in terms of when benefits begin relative to your start date.
If it turns out that you would be left with a gap in an important benefit - such as healthcare - you still have options. Namely, COBRA regulations give you the option to elect to continue under your current employer's healthcare plan.
Essentially, COBRA gives you from 18 - 36 months of access to your current plan, and you have up to 60 days to elect that coverage. Further - and importantly, in your situation - you can elect the coverage retroactively back to your termination date at any point during that 60 day window.
The catch to COBRA is that it is usually very expensive, since the employer will typically elect not to contribute - so you will likely be paying a significantly higher premium.
This all basically means that if you end up with a gap in your coverage, you can simply wait and see if you incur any medical expenses that would be covered. If you do, and those expenses would cost more than the COBRA premium, you can reach out to your plan and elect coverage back to your term date, and the expenses would be covered. Or, if you already know you will have expenses during the time in which you'd have a gap, you can just elect coverage upfront. Once you have COBRA coverage, it behaves just like your plan had in the past, in terms of claims, customer service, and so on.
So - if your ultimate concern is losing healthcare coverage betweek jobs, that's likely not an issue, even if your employer changes your final date.