English Teach Away in Singapore--does this exist?

All schooling in Singapore is entirely in English, so there is very little demand for TEFL. Those  few jobs that exist are filled by local freelancers. There is no visa type for freelancing and, without years of TEFL experience, you won't qualify for any such job anyway.
There is of course also no “standard job for American college graduates in Singapore". (What were you thinking of?)
But why are you writing off your biology degree? Singapore is great on research in genetics and bioengineering. But of course to qualify for a job, you normally need about three years of applicable work experience. So you better get that first!
Alternatively, marry him and move here on LTVP (provided he has sufficient income). After three years (or with a male child) you can become PR and then you are allowed to do freelance work.

Yes, government schools conduct classes in English, and most local kids speak English at home, though there are some foreign and some local families who primarily speak another home language.

However, actually there are many companies that offer English phonics and reading, writing, listening and speech & drama for children as supplementary "enrichment" classes that children attend for say, 90 minutes once a week. Some hire foreign native speakers of English, especially but not necessarily only those who have a background in linguistics, English literature, or education, and have a CELTA or TEFL certificate.

Google "English enrichment for kids", "English tuition for children", "English speech & drama", "phonics lessons", or "creative writing" as if you're a parent looking for these kinds of classes, or browse the lists of businesses at:
https://www.kiasuparents.com/kiasu/directories/