There is a chance you could get it in your throat, and then transmit it to someone else you give oral to.
You should go to Planned Parenthood or your own doctor and get an oral chlamydia test - make sure they know to do a specific chlamydia test, as a regular throat swab usually doesn't test for chlamydia.
They will also probably do a gonorrhea test at the same time, but those are normally run together.
You can wait to tell your 2nd partner after you get your test results. Getting chlamydia from oral sex is less likely than getting it from intercourse, but it is possible. The treatment is the same, though. It's very easily cured with antibiotics. If you've had intercourse with either of these partners, you should get a vaginal test, too.
Also, don't forget that covid is still a thing in most places, and if you're having oral sex, you aren't social distancing. You might ask for a covid test as well, especially if you're in a high covid area.
Let us know what happens with your tests.