First of all, the fact that they approached you is a good sign. Despite the strong positions I recommend below, be positive and cooperative. These seem like people you want to work with. My thoughts, some redundant to above but with different spin:
1. Do get an experienced lawyer, but ask the money guy to pay his lawyers to draft the documents. That is the expensive part. You say the technology has multiple uses, so be careful, when granting them exclusivity, to define their reserved market.
2. Look at the organizing documents for the new company and understand the relationship between the spinoff and the parent company.
3. Include terms for both the parent and your subsidiary in your license, see 2 above.
4. Address how the patents will be enforced if a competitor uses the technology. If you are granting exclusivity in the market, are you obligated to enforce the patent to assure exclusive use? Will the startup have the resources and the will to enforce the patent?
5. Regarding your role, since participation is important to you:
a. If the new company is built around your technology, you might ask for a seat on the board.
b. Negotiate an employment contract coincident with the license agreement. What do you want to do, where do you want to work and who do you report to.
c. If they believe in your technology and the enforceability of your patent that should be reflected in their offer for the one time exclusivity payment and royalty rates. Since they approached you, it appears they believe.