You should have no problem finding a reliable host with your budget, Have you searched the offers section.
GS RichCopy 360 Enterprise - Voted #1 for data migration and replication in terms of performance and features. Replicate data across between servers in the same network, WAS, or even across the internet
Get a list of providers and search for reviews of them. You can check them at http://www.whreviews.com or http://www.hostjury.com and even try running the name through Google. Your budget seems right on target and you really shouldn't have any trouble at all finding reliability within that range. Good luck!
$10 - $15 a month is low considering you want 5GB with multiple domains. You may need to increase this above $20 per month to ensure you are getting the necessary features such as RAID 1, nightly backups and support.