The types of wart virus (human papilloma virus or HPV) that cause ordinary skin warts are different from the ones that cause genital warts. So, yes, it is true that transmission of HPV can occur with a handshake, but it is unlikely that you could get one of the types that cause genital lesions. The transmissibility of HPV to intact skin is very, very low. Here is a table of types of HPV.
Treatment options are well described at hpvinfo.ca.
Our bodies are already protecting themselves. If you have a break in the skin of your hands, you are likely to cover it up. Handwashing is a good idea for lots of reasons, and will decrease the risk as a side effect, but I would not increase my frequency of handwashing because of this minor concern. There are many threats to human health of far greater magnitude than getting warts. Protect yourself by wearing seatblets, exercising regularly, and eating lots of fruits and vegetables. Ignore the extremely tiny risk of getting ordinary warts. It's not worth the time.