You don't need static ips for doing sharing. Share should be setup as servername\sharename so the "server" can have any dynamic ip address and will still work. This also means unless the share was setup by ipaddress\sharename you would not need to remake the share if you changed the router and that new router was providing a different ip subnet than the old router.
Either one of two things are happening while booting up and trying to reconnect the drive mapping;
1. your password has changed on the local drive and does not match with the password your account has on the mapped drive
2. you were unable to contact the shared computer
1. reset your accounts password on the shared computer to match the one you logon with to your computer
2. make sure the shared computer is on and connected to the network