I think I got frostbite. Is my toe going to get cut off?

Watch and wait. It is very difficult to estimate the amount of tissue that will necrose (die) as a result of frostbite. Conservative treatment with gentle rewarming and observation is typically recommended. Removal of dead tissue is performed only when it has fully demarcated.
Avoid infection! Obviously, follow your doctor's advice as he or she can see the wound, but silvadene (silver sulfadiazine) cream can help prevent infection and decrease the amount of tissue lost. I hope things go well for you.
Hopefully not. Get seen asap. Gentle rewarming is the best treatment after you can be sure no further cold damage will occur. Any surgery is done only when the depth of the injury can be fully judged (i.e. Not right after rewarming).