[gnrx]

I currently manage my site in a VPS that of all those I've tried in the 14 years of my site's life, is the best by far, good hardware, good price ... but I think it exceeds your budget.
Any VPS will always be more expensive than a shared server (which is what you have now).
In any VPS you can select which version of php to use, and in many shared servers, too, you don't know that dreamhost doesn't allow it.
A VPS is like a dedicated server, but it is virtualized. You don't need the resources of a dedicated, but it is, like a dedicated.
In which I am, I am delighted, that yes, it is self managed, I mean, you must do everything, whatever you need to change on the server / VPS you must do it. It's not like others who give you the service to do everything (called full managed) but of course, the bill at the end of the month is higher.
Don't worry or be scared, administering a VPS is not that difficult, and there are guidelines for everything.
If you are still interested, I can tell you where I am, but you go to 16.99€ per month (this includes the VPS and the cPanel license).

Greetings.