No, your address is your account. You can't change it. You need a new account if you want a different address.
Gmail invitations are available without an invitation, but currently, only to cell-phone users in the US. But invitations are relatively easy to get (particularly in forums like this one.
)
But if you'd prefer to not have to seek one, use your current Gmail account to send yourself an invitation. It would, of course, have to be at an address not currently associated with your account.
Use an alternative, if you have one, or get a temporary free account like Yahoo to send the invitation to.
Oh, and I wouldn't bother cancelling your current one. It could come in handy to use in situations (online registrations, etc.) where an account needs to be potentially revealed to spammers.