As for the HSA, you are eligible to contribute to the HSA for any month for which you had HDHP coverage (and no health FSA) on the first day of the month. To be precise, the calculation of your annual HSA contribution limit will be the full annual limit times the number of months you had HDHP coverage divided by 12.
Thus, if you had HDHP coverage from January 1st to August 15th, your annual HSA contribution limit would be the full HSA contribution limit for your HDHP coverage (Self or Family) times 8 divided by 12.
The FSA question is a trickier one, because it depends on how your FSA plan is written by your employer.
For 2020, the maximum amount you can contribute to a health FSA is $2,750. However, one of the rules of an FSA is that you can be reimbursed for the total amount even if you haven't contributed it yet (i.e., the employer covers your expense until your contributions catch up).
If your FSA plan prorates the maximum contribution to your health FSA based on the number of months you had the FSA coverage (in your case, perhaps September to December), then the most you could be reimbursed is $916.67. In this case, you would not be able to take advantage of both plans in an overlapping manner in the same year.
But if your FSA plan is written to allow you to contribute the maximum $2,750 for the year even in four months, then you would have overlapping benefits.
I don't find an IRS Notice that describes this (and the instructions certainly do not), but my thought is that if your employer's FSA plan limits the amount of your maximum FSA coverage in your partial year, then you should be OK. The problem is if your FSA allows the full $2,750 for the partial year, then that would be as if you have FSA coverage all year, which would disqualify your HDHP coverage.
Ask your HR department about the particulars of the FSA plan.
