If you're disclosure was documented properly, it would prevent them from "stealing" your idea and patenting it. It doesn't necessarily prevent them from using your input to think of a similar product and patent that product.
Strictly speaking, it doesn't matter whether your disclosure was "documented properly", only that you disclosed it to them. On the second point, the "similar product" would have to meet the requirements of section 103 in view of your disclosure to be patentable, something that may be difficult for something "similar" to your idea.
If I were you I would at the very least file for a provisional patent before even considering the disclosure of the idea to anyone unwilling to sign an nda.
A provisional may be a good idea, but a properly drafted provisional isn't necessarily all that cheap. If done properly, IMO, it should cost the same as a formal application minus claim drafting time. Market place realities are such that big companies often won't sign anything.
- Jeff