I\'m setting up a webserver for a system that needs to be used only through HTTPS, on an internal network (no access from outside world)
You have two practical options:
Stand up your own CA. You can do it with OpenSSL and there's a lot of Google info out there.
Keep using your self-signed cert, but add the public key to your trusted certs in the browser. If you're in an Active Directory domain, this can be done automatically with group policy.