I've seen mixed information online about how oral gonorrhea is transmitted. Is her advice correct in that I should abstain from more casual contact, such as open month kissing or sharing food/drink?

It is rare that ceftriaxone 500 mg fails to cure pharyngeal (oral) gonorrhea, but not unheard-of; probably it happens 1-2% of the time. It does not necessarily mean the gonorrhea is antibiotic resistant: some (probably most) oral treatment failures occur even with fully antibiotic sensitive strains; and repeat treatment with the same dose usually is effective. I agree with the advice you received -- both about retesting again in two weeks, and in taking precautions to prevent any chance of transmitting your infection to anyone else (but not with advice to avoid sharing drinks or food).

For sure you must not perform oral sex on anyone, and it makes sense to avoid sexual (open mouth) kissing as well. There is controversy about the possibility of transmitting pharyngeal gonorrhea by open mouth kissing, but probably it happens once in a while, so please don't take the chance. Oral gonorrhea is not transmitted by sharing food or drink, or by social kissing (on the cheek, peck on the mouth). It would be best to avoid other sex as well -- in theory vaginal sex should be safe, but given natural tendencies and some partners' expectations, it might be difficult to have intercourse while avoiding kissing. Best to just plan on no sex at all for the next couple weeks. But definitely no worries about normal non-intimate contact, including the possibility of sharing drinking glasses, eating utensils, and social (hello/goodbye) kissing.

I hope these comments are helpful. Let me know if anything isn't clear.

I did have a follow up question: I reused an old toothbrush while I was still infected (about 2-3 days old) this evening. I did wash the toothbrush in hot soapy water for about 10-20 seconds before reusing it. Am I at risk of getting reinfected?

The toothbrush isn't a problem. Neisseria gonorrhoeae -- the gonorrhea bacteria -- is very fragile and could not have survived on your toothbrush.