Ken (and others), you might have some thoughts about the solution that I have right now.
Well, many folks do not like GoDaddy. However, GoDaddy is the largest hosting company, and they probably know more about security than any school's IT staff. So from the server side, I would find it hard for someone to argue that Godaddy's servers are "not secure." There are probably many other good hosting providers with equal security.
Then, it's just on the Moodle side where you need to take care of things, following some of the suggestions already made.
Cost: Depends... but figure $20US - $30US/month for a VPS. I pay this myself, not my school, and I find it within my budget for what I seek to do in my courses.
This VPS is one that you manage, and you have a lot of control over the components. For example, mind is running php 7, MariaDB, Opcache, and I have optimized my database and php settings (php.ini and my.cnf).
Well, takes a bit more education. Nothing too major.
Yes, you can move your current moodle to a VPS (if you know what you are doing.)
I do not own stock in GoDaddy nor do I have any affiliation with them. I have used GoDaddy for about a dozen years. My experience is that their servers do not go down. Oh, one more thing... GoDaddy does not provide Moodle support (nor MS Office, nor PC, etc.) Their business is servers.