When you are a contractor you don't really have a boss. You have clients, you need to start getting yourself in this mindset, even though the end result ends up being more or less the same.
So again, no boss, but you have a representative from the company that you have a contract with that is your liason and the person that is giving you the work. You need to keep them happy like a boss, but they are not your boss.
You don't get overtime from the company because you are not an employee of theirs, you are employee of your own contracting business. It is up to you whether you pay yourself overtime or not.
Whether there is money available for this or not in the contract depends on how you negotiated the contract.
Regarding not being able to work for other clients, this is just not true. You are your own business and you can take on all of the clients you please. The documents you signed are meaningless and do not hold any water. Of course if you do take on work for other clients, you do not need to let your current customer know unless it will materially impact them in some way.
> I work on a team where everyone else is an employee and I don't seem to be treated differently.
This is normal for contractors. You work alongside regular employees and everyone is focused on the same goal. The only issue is that you need to not think of yourself like an employee. You work for yourself and you create the rules around how you operate your business.
I think what you need to do more than anything else is to talk with other contractors about how they deal with clients.
Get better information about how long they take contracts for, how many contracts they take at once, how they negotiate prices with prospective employers and what formulas they use to factor in external costs, ask them how they budget the money for their operations, how far they plan ahead, ask them how they do their own healthcare.
You need to take your business to the next level, and currently you seem to be allowing a customer to drag you around by the shoestrings to do whatever you want.
It's time to start setting some better rules around how you operate, instead of just going with the flow.
I believe getting this advice from other people who are currently do this is probably the best way to start.