I used to use Hostgator shared hosting, had a few issues like this and got in contact with them, they assessed the sites on the server to see who was hogging the resources, after about 30 mins it was fine, not sure what they did.
I'd open a ticket anyway and get them to check it out.
I don't think you need a dedicated server yet, maybe a VPS.
Check out futurehosting.biz. I had a VPS with them and don't have a bad thing to say about them, very professional company.
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