The word "theorem" implies that there is a proof. If you do not have a proof, you do not have a theorem. Let's call it a conjecture instead.
I have tried disproving my theorem for over a year.
That's a slightly strange thing to say: how long have you spent trying to prove your conjecture?
Should I just publish it without a proof, Can I even publish without a proof?
It is possible to publish conjectures, but it is significantly harder to do so than to publish theorems. I would not recommend that a young researcher -- especially, a high school student! -- try to do this.
I have contacted professors and have conversed about this and they all say that it is definitely publishable, but that I should prove it myself.
It's no help if I'm not honest, so: this sounds fishy to me. It is often not so easy for (even) a professional mathematician to know what is "definitely publishable": after more than a decade of submitting math papers I find that I still have some things to learn about this. But anyone who is telling you that your mathematical work is "definitely publishable" if it does not contain a proven theorem is either giving you bad advice, or you are misinterpreting the advice.
How can I ensure that my ideas are not stolen, Can I provide direct proof that the idea is mine somehow?
With probability extremely close to 1, professional mathematicians simply do not steal ideas in the manner you are worried about. The "I did something great, but oh no I can't show it to anyone" train of thought is a sad one that amateurs often fall into. Rather, if you do something great, show it to more than one person, and you're fine. One way to do this is just to upload it to the internet in some public or semi-public location. For instance, if you have an account on facebook, just post a scanned copy of the paper as photos. Facebook posts are archived with date and time, so that's that.
I will extend to you the following offer: after you archive your paper publicly on the internet, send me a copy. I will spend up to one hour looking at the paper and tell you one of the following:
- The mathematical content of your work is such that you should try to publish it. I will then tell you some places you might send it.
- In my opinion your work is not publishable in a reputable, professional mathematical journal.
- I am not qualified to judge whether your work is publishable, but I recommend that you send it to third party X.
You should understand that unless the work is very directly connected to my own I will not have the time to help you with it, nor to send more than one email. (I simply can't: I have a lot of other people who are counting on me to spend my time on them. In many cases I am being paid to do so.) But I will give you a professional evaluation of your work so you can (probably) know where you stand.