1.) Learn about other wallets, such as Exodus (https://www.exodus.io/). Exodus does not hold all altcoins, however. You might be able to get more altcoins with Exodus Eden (https://www.exodus.io/eden/).

2.) What others have done () is have two MEW accounts. One MyEtherWallet account is used to temporarily hold altcoins and is known by your exchange(s). You keep a second (or third, or fourth) MEW with the majority of one or more altcoins, but never use it with an exchange - only the first wallet is ever known by address to an exchange. This may help deny hackers.

So the answer really is "get more wallets, and don't hold all your crypto currencies in one wallet".