Yes, you can date someone else after you separate from your spouse. There is nothing illegal or wrong about dating while married and waiting for your divorce as long as you are living separate and apart.
Is it adultery if you are separated?
Lots of people are able to handle the messiness of sex with a separated spouse; other people find it to be emotionally difficult. As long as youre married, having sex with someone who isnt your spouse counts as adultery. If youre separated at the time however, no one except your in-laws or the Pope is going to care.