You also asked "And what will be the contribution limit for both HSA with my old employer and FSA with my new employer?"
The contribution limit for the HSA was already discussed.
The contribution limit for the FSA is $2850 for 2022, the 2023 limit has not been announced. Your FSA limit is not reduced because you contributed to an HSA in the same year. But remember, an FSA is use-it-or-lose-it. Don't contribute more than you reasonably expect to spend during the plan year.
Also, be sure to know what the plan year actually is. With my current employer, the plan year is July 1-June 30, but most employers will be January 1-December 31. You have to think about how much of your 2022 plan year will be left when you take up the new job, so you can plan how much you might spend in medical costs during that remainder.
Also be aware if your plan has a grace period or a carry over. An employer is allowed to offer a grace period of up to 2.5 months to spend your unused FSA balance. That is, if your plan ends on December 31 and you have an unused balance, you can spend it on medical care up until February 15. Or, you can have a carry over of up to $570. That is, if you have an unused balance at the end of your plan year, you can carry forward up to $570 to the new plan year. An employer can only offer a grace period or a carry over, not both, and an employer doesn't have to offer either one.
Once you know the plan year remaining and the carry forward or grace period rules, you can think about how much you should contribute.
