Basic misunderstanding, you coins are always values contained on the BTC blockchain. Your wallet only keeps the keys and reports what is on the blockchain for those keys/addresses. The private key allows you to create a transaction for the blockchain that the miners will approve. So you do not need to move anything anywhere to use a new wallet.
omni-assets are stored in the BTC blockchain so the above applies to them.
You should see MAID for the address you imported the private key to.
That would suggest that you didn't import the private successfully. Although the omniwallet should not be showing the address then. (unless it is a watch only address)
There is another topic here that discusses the issues with importing private keys. Omni-wallet can only import private keys starting with "5" even though it says it can do more.
Read this, it might help