How much CPU time did you use? I’d love to know where the limits lie. Theoretically they are supposed to work with you and put your site on an underused server, but maybe you reached those limits as well. How many CPU seconds were you using per day? How many pages served per day?
As for the VPS, they start you out with max RAM, and you probably don’t need that much.
If you can’t afford a VPS (dedicated server would be more expensive), then look at optimising your code and going back to shared hosting. Is your site based on PHP? Does it have any caching routines?
Another option is to form a non-profit and apply to for a free account. DH hosts non-profits for free.
Also, you could try cloudflare, or is it cloudfire?, which is a free service and supposedly takes some of the load off of your server.