Is using 'Cookies' impractical for 'Contact Us' forms?

What about those folks whose computers don't allow cookies to be enabled (e.g. their PC at work)? How to allow those folks to use the E-mail form to 'Contact Us'?

Posting phone#'s is getting abused at times in similar ways as mining E-mail addresses for mailing lists. I see more and more websites having E-mail-only (via E-mail forms) as the only method for contacting them.

I want to use cookies for session management where the cookie expires. This includes image verification that uses cookies. I'm trying to create some resistance to bots; multiple submissions; protect website in shared environments; etc.

Is this a Catch-22? Do I have to decide to just not have everyone with the ability to 'Contact Us'?

Currently, I am setting up to use cookies with a database for sessions validation, but am I locking out too many people if I use this method of security, thus making it 'impractical'?

Should/Could I use another method instead of cookies, e.g. a file that is deleted after some time period which would contain the same info as a cookie and still allow for comparing the session entry stored in a database? If so, how?

Thanks in advance.