Changing working IMAP folder doesn't require you to reconnect. Simply issue new Select command:
using (Imap imap = new Imap())
{
imap.ConnectSSL("imap.example.com");
imap.UseBestLogin("user", "password");
CommonFolders common = new CommonFolders(imap.GetFolders());
imap.SelectInbox();
List<long> uids = imap.Search(Flag.All);
// do something with uids here
imap.Select(common.Drafts);
uids = imap.Search(Flag.All);
// do something with uids here
imap.Select(common.Sent);
uids = imap.Search(Flag.All);
// do something with uids here
imap.Close();
}