If you have no symptoms, you can have sex at 7 days post treatment. This includes any sores, any rashes, unexplained headaches, fevers.
Make sure that your 3rd partner has been tested for all STDs, including syphilis and HIV, before including him in your sexual activities. It would suck if you were cured, only to be reinfected.
Also, if you're in a high covid area, talk about what kind of steps you've all taken to reduce risk. Sex doesn't transmit covid, but close contact sure does.
You should have been told that you'll go back for a follow up blood test. That might be in 3 or 6 months (this is where that time frame you saw comes in.) At that point, if your titer hasn't dropped 4-fold, in your case to 1:4, you will probably be retreated. Until you and your partner get that confirmation of cure, you might want to use condoms.