Is there a sent folder . . .?

[rgG]

I really don't want to use a new account if I don't have to.

Is there a way to search all mail when needed?

Yes, if you click on the "inbox" then put a word or phrase in the "enter keywords" in the box at the top and then hit "search PMs" it will bring up PMs with the search criteria, even though they may not show in the inbox. You can also do an advanced search to narrow down the responses. 

HTH