I became aware of my skin problem I think a few weeks after kissing a girl. Although it could also be before it, I thought about it many times, but I can't recall whether I had the skin problem before it, because at that time, I didn't took it serious, just like it was some form of acne. But if it was caused by her, then it's caused/transferred by kissing or skin-to-skin contact. But if your current partner doesn't have any problems at all, than that would make that unlikely again, or they are asymptomatic.
I have also used doxycycline, it did make my skin smoother in general I think, but it didn't remove the oozing lesions at all.
Also note that chlamydia is often tested by taken samples from the genital area, such as urine or swabs. But if it's located on another area, then I don't know whether samples from the genital area would detect it, probably not, unless maybe if the test would be done via blood.
I did do a STD test to rule things out for the lesion disease, which came back negative. But if Chlamydia is capable of being on a different location, then it may be a false-negative. I find it interesting that you both mention this.