Since you are not using a control panel, there's no other better way to do this.
I suggest that you try Webuzo on a new vps and transfer the sites one by one and test before switching nameservers.
If you can't do it yourself, or if you are not confident, you should find a freelance Linux SysAdmin to help you. It might cost you $30 - $80.

I wouldn't be confident with transferring the databases for the website etc, in case something goes wrong.

If you take a mysqldump and restore it (verifying the hashsum first and transferring it securely via scp), you should be fine.

Honestly, I don't recommend installing zPanel (or any other control panel, really):

I have a couple of websites that have been set up through apache virtual hosts, and it is serving my website fine.

Since it's working well, why do you want to add another layer of complexity and install zPanel?

"... and it is serving my website fine."

Why are you satisfied with "fine"? Would you not prefer optimum performance? Without knowing more about your site, I cannot say for sure; but Nginx might be a better option (box, Netflix, SoundCloud, Facebook, hulu, Zappos, Groupon, Zynga, Intel, Eventbrite, WordPress, Dropbox, DigitalOcean, Cloudflare have all dumped Apache in favor of Nginx). In July 2013, Nginx became the most-used web server among the top 1000 web sites.

"... I want to be able to manage it more effectively..."

Who told you that ZPanel is the answer? A cursory review of the DigitalOcean support threads shows that A LOT of people struggle with ZPanel (and many times, even with the most elementary of functions, i.e. DNS records).

"... and I wouldn't be confident with transferring the databases for the website..."

Then NO control panel is for you, my friend. As Kamal duly noted: Why add yet another layer of complexity to your web server?

"I guess the question really is, what is the best way to go about this?"

Read more DigitalOcean guides. Also, there are TONS of (free) video tutorials on YouTube to help you learn more about Linux (I subscribe to the Eli the Computer Guy channel).

Stick with the command line. Control Panels' true purpose is to simply provide a graphical user interface (GUI) to administrative functions. But, you still have to (or should) know exactly what those underlying control-panel-functions are doing and what the end result will be.

Just get a new droplet and set it up. Once you get the control panel installed on the new droplet with working users, transfer the files over.