Yes, leave your prescription meds in their original containers, although I don't believe that you need any kind of note from your Dr for domestic flights.
But I agree with the above: Never ever never ever never ever never put any meds in checked-in luggage! You really are asking for trouble by doing that. Besides, regardless of lost luggage, if your plane was significantly delayed or re-routed and someone needed those meds, you wouldn't have access to them.
Definitely either put the meds in the safe in the room, or put them in an inconspicous place, such as a sock drawer.
My son has special needs and takes several meds. We have done a lot of domestic travel and have never had an issue with the meds. In fact, I normally carry them all in my purse for the duration of the trip, along with Motrin, Tylenol, etc... You just never know what type of situation may arise that could keep you from getting to the meds.
Good Luck!