Sorry just seen my typo I meant ports 143 and 993.
Irrelevant. Take a step back and think about it logically. The error messages did not say that you could not connect to these ports. In fact, it seems you can, so clearly, this is not your problem.
A "netstat -nat" will show if something is listening on these ports.
doveadm(root): Error: Timeout waiting for handshake from auth server. my pid=8881, input bytes=0 doveadm(root): Fatal: Couldn't connect to auth socket
This is what you should be looking into -> "Timeout waiting for handshake from auth server"
The second message "cannot connect ..." is the generic error if the entire connect fails. However, the primary error is the timeout error. That is the key. It is not getting a response from the other end of the socket. What is supposed to be connecting to the other end? Are you using SASL or something else to do authentication? What does "lsof | grep auth-userdb" show? Two things should have that file open. If not, there is your problem. A pipe, named or otherwise, is like a chatroom. It is very lonely when you are the only one.
Standard problem solving techniques. All the information you need is right in front of you.
Dem