I wonder whether, when applying for a new US passport at a Passport Acceptance Facility (e.g., some USPS or county clerk locations) via Passport Application Form DS-11, I can go to any Passport Acceptance Facility or does it have to near my domicile (e.g., in the same state or USCIS district).
You can go to any facility.
The actual passport issuance in the US is centralized and all the passports are issued by the State Department (as opposed to other countries where you may see "Issued at city X"). The Federal government doesn't care where you live. Notice that the address field on DS-11 is for mailing address. They ask for your residence address (on page 2) if it differs from mailing, but it is not a criteria for passport issuance.