Excellent analysis given little info. Although it is possible the VPS host is not managing its resources well and its possible Apache is configed poorly, it is just as likely the app itself is a pig and needs lots of cleanup.

If you don't have the skill set be a good sysadmin and you "dont know" if its your app, you are in for a world of hurt. I highly suggest you find help: a good sysadmin and maybe an outsider to look at your code and web page structure. I have not heard enough to say your hosting company is screwing you. And its maybe wise that you are paying $60/month for better tech support but you still only get $60/month worth of tech support and that's not much ;(.