You are not the only one experiencing problems and I don't think it has anything to do with the Fantastico installation either. I have a relatively large non-Fantastico Moodle installation on LunarPages and we have had nothing but problems during the last couple of weeks. We get database connection errors etc, when there is not that heavy a load on the server - we are still in the process of loading students for the new semester. I have no idea what is going on but something is very wrong somewhere. Our resource usage stats show relatively large amounts of CPU and memory usage. I find this odd when I know that there's not that large a number of people accessing it. I have also checked the folders for any sign of a hack but can find none so there are no strange, rogue scripts running. I'm baffled and just a little fed up.

I'm not sure how much experience you have with Moodle but rather do a manual installation without using Fantastico. For one thing, Fantastico places the data folder in the Moodle folder which is not at all advisable. In fact, the data folder needs to be in the root folder.
I'm not sure if you are able to set up a cache accelerator such as eAccelerator on a shared hosting plan. I'm sure the support desk could help with that.