Hello,
Warts are contagious when in contact with the skin of an infected person. The strain of virus causing genital warts is different from the one causing common warts but if visible sores were present on tha hands of your partner then HPV can be transmitted. It is also possible to get warts from using towels or other objects used by an infected person.
But warts do not show up on the same day after sexual exposure. They may show up after some time.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.