Finish previous work during current position
As an academic working with a couple of post-docs, my rule is you can do what you want with your time as long as you do what is required for the project that employs you. 
It is also the case that I know (and indeed expect) that post-docs will have work of their own that they'll want to finish up or extend.

I would not be happy with option 1, but option 2 would be good for me. Weekends are your own. If you spend them writing more papers, I will not object. But please do have some rest and relaxation time.

As a post-doc, you are starting to build your own career, so writing independent papers, and thereby forging your own way,  is a good thing. But there are a lot of benefits to be had from integrating yourself in your new working environment, and working too much on your own stuff means that you will miss that opportunity (and possibly make a bad impression).