Well, goodness, doxy is taken for all kinds of reasons - acne, bronchitis and other upper respiratory infections, eye infections, gum disease, even malaria. It's a broad spectrum antibiotic, which means it's used for many, many things.
She's a CSW. You aren't entitled to her medical history. This is not a relationship where you share personal details.
Oral IS a risk for gonorrhea, chlamydia and syphilis. Since she's on doxy, apparently, you can rule out syphilis and chlamydia. Gonorrhea would be the only risk, and I wouldn't worry about that. Doxy would probably prevent that, too.
Just a note - if you are adult enough to have sex with CSWs, you are adult enough to have tough conversations.