Can you also get chlamydia from not washing your private area after unprotected sex?

Theoretically: yes: Chlamydia is trransmitted in sexual secretions so: if you had sexual secretions near or in your vagina guess what?- your are officially at risk and.....time to rethink the constant use of condoms. Please consult your GYN if you are having a discharge and ALWAYS consider how much protection you have when sex is with condoms.

Unprotected sex: will put you at very high risk of acquiring Gonorrhea, chlamydia even if you wash your private part. Practice protected sex to reduce risk of STD or unplanned pregnancy

Yes if your partner: has chlamydia, and even if you do wash the area. You don't want to engage in unprotected sex unless you are sure your partner is free of STD's. And if your partner is clear of STD at the moment but has a history of genital warts or herpes, you can still get the virus even if your partner doesn't have lesions. You should have your partner checked and see their results and wear a condom.