A VPS is a little more involved when compared to shared hosting. The main differences being that in comparison, you would be responsible for setup, maintenance, security, and overall management of the VPS and OS -- as well as optimizing the stack to meet your needs (performance or otherwise).
A VPS will provide you with the freedom to scale, and set your own limits, so you would definitely be able to get around the memory limit.
Most shared hosting providers use cPanel combined with CloudLinux, which is where the limitations are coming from. CloudLinux allows the web hosting provider to limit by CPU, RAM, and Disk I/O, so when your account hits the ceiling of these limits, it's either slowed down, or errors out.
That being said, you would most likely be better of on a VPS. Since you have 10 WordPress sites and the current limits are too restrictive, then I would recommend starting out with at least a 1GB Droplet since you need room for Apache or NGINX, MySQL or MariaDB, PHP, and your sites (all of which are using resources such as CPU, RAM, etc).
My recommendation, however, would be to go with a 2GB Droplet which should provide a little more room to breathe.
Setup isn't really all that difficult and there are multiple guides here on DigitalOcean that can help with the setup -- there's also the community which I and others are quite active in, so if you run in to any issues, we can normally help get you squared away.
One thing I will make note of offhand is that unless configured otherwise, a VPS (Droplet) doesn't provide a working mail server. That'd be something you'd have to configure on your own if you need e-mail functionality.
In most cases, I recommend offloading e-mail to Google G Suite or a similar provider as managing a mail server and keeping your IP's / Domains off blacklists can be a pain.
How To Install Linux, Apache, MySQL, PHP (LAMP) stack on Ubuntu 16.04
How To Install Linux, Nginx, MySQL, PHP (LEMP stack) in Ubuntu 16.04
The first tutorial is setting up a LAMP Stack -- that'd be similar to what you're running on now as it's using Apache, MySQL, and PHP. The second tutorial is for NGINX, which is more performant and in my opinion, easier to work with and manage than Apache.
My recommendation would be to go with NGINX exclusively -- I'm a bit biased as that's the web server I work with the most, but I also recommend it simply because I find it to be more performant than Apache and it handles resources better whereas Apache be a resource hog at times.