It's safe to use plastic. We do it pretty often, and never had a problem.
For small amounts we just use dollars. There is no need to use Mexican currency.
We have been to Cozumel many times, and it's one of our favorite ports.
Carnival Liberty New Year's Eve 2007
Liberty of the Seas 5/2008+11/2009
Allure 1/16/ 2011
Vision OTS 2-14-2015
Freedom OTS 3/6/16