There is no problem to use your web hosting account at AvaHost.Net with more than one domain name.
Alternatively, if you don't want to have separate websites but just separate domains which load the same website, you can park your existing domain names to your primary one. You can park your domains from your cPanel > Parked Domains (Watch movie)
If you are having problems using your account with more than one domain at your current host, you can sign up for our cPanel hosting package. AvaHost's support team will happily assist you with configuring your account to work with more than one domain.