Is there a way for me to avoid paying Coinbase fees?

There isn’t, not if you want to continue using Coinbase.

You also can’t escape paying miner fees for any transactions you do on Bitcoin, or any other crypto network for that matter. BUT you can save drastically on fees by using your own wallet client. For Bitcoin, examples would be Bitamp wallet or Electrum. Both interact directly with the Bitcoin network, so you pay only and exactly the required miner’s fee to confirm your transaction.

You will save a lot in the long run. As an example, a withdrawal fee on a typical exchange is at least 50,000 satoshi today. But sending your own transaction from your own client will cost you as low as 150 satoshi. That’s 300 times more if you use an exchange!