Where is the abscess?
If it was a Bartholin's gland cyst, yes, it's possible.
https://pubmed.ncbi.nlm.nih.gov/32321682/
"64 patients had a diagnosis of Bartholin cyst or abscess, 40 of whom were also tested for Neisseria gonorrhoeae or Chlamydia trachomatis. Ten percent of patients with a Bartholin cyst/abscess were infected with N gonorrhoeae, compared with 3% of those without a Bartholin cyst/abscess (P = .008). The rates of C trachomatis and Trichomonas vaginalis infections were 13% and 26%, respectively, among patients with a Bartholin cyst/abscess, compared with 8% and 30%, respectively, among those without a Bartholin cyst/abscess (P > .05 for both). On regression analysis, only increased urobilinogen level (β, 0.31; odds ratio, 1.36; 95% CI, 1.11-1.66; P = .003) and infection with N gonorrhoeae (β, 1.69; odds ratio, 5.40; 95% CI, 1.43-20.35; P = .01) were associated with a Bartholin cyst/abscess."
It's a small study, but revealing.
https://www.healthline.com/health/bartholins-abscess
If it was a different type of abscess, let me know.
Draining an abscess quite often immediately relieves the pain. It's very important to follow the care instructions you received. Infection with other bacteria is quite possible otherwise.
Your husband needs to be treated for gonorrhea as well, as well as other partners either of you have had in the past 3-6 months.