All of my website service pages have a FAQ section with different questions and answers for each page. Would the FAQ schema markup be eligible here? Or does it need to be a single FAQ page?
Based on schema.org FAQpage definition
A FAQPage is a WebPage presenting one or more "Frequently asked questions" so there is no mention that the FAQ page must be a dedicated page, separated from the content it provides answers for.
On top of that, you can (and most sites do) use multiple schemas in the same page to describe the different data that are presented on it, so it is safe to use the FAQpage schema alongside any other schema that page already has, without worrying that the FAQ section should be in a separate page.
Schema.org doesn't make suggestions, how to design your internal architecture. But it supports any architecture.
I would design FAQ sectors so, as it is easier to your visitors to get their info. Let usability guide your design. Schema.org allows pretty everything.