At 16 you will not be able to find accommodation in hostels, and couch-surfing at that age is illegal - both for you and the person who offers a minor accommodation. So forget that idea.
The exchange organisation is responsible for finding you somewhere safe to stay during that period. Dong your own thing is a big deal, in fact it's a major breach of the rules. Breaking the rules will result in you being sent home - I've seen this happen with exchange students. And it will be a huge disappointment to your parents who have invested a lot of money in this trip on your behalf.
Do not do it.
At 16 you may have difficulties getting the tickets and a place to stay, as more and more hostels and ticket sellers check the age of teens and do not accept if under 18 and without the proper permissions of parents (or exchange host).
Couch surfing is less risky than staying with a stranger you meet in the train but it is still not safe, certainly not as young female.
At 16 you will not be able to stay in a hostel or anyplace like that by yourself, as you cannot legally sign a contract.
As the program is responsible for you and you are a minor (USA), it could get you sent home. That is not a mark you want on your school record. You could with your parents permission stay after the program and go to Amsterdam.
Sounds dangerous, but if you're into being a lone adventurer like I am I say go for it, but you being a female will make you more of a target, stay alert