The good news is, there are effective strategies for dealing with these birds. Although, none are without trade-offs.
Option 1, Switch to Safflower exclusively. Birds like Grackles and& Starlings can’t eat safflower, but … Cardinals, House Finches and Doves love it! Another benefit is that Squirrels don’t like safflower either. You can switch back to a mix in the winter when most Grackles migrate south.
Option 2, Use a caged tube feeder or “clinger only” feeder. Caged feeders only allow small birds access to the feeder. If you use a caged feeder and would like to feed larger birds like Cardinals and Doves, you could offer safflower in another easily accessible feeder. A clinger only feeder is a feeder with out perches. Birds like Gold Finches, Nuthatches, Chickadees and Woodpeckers are considered clinging birds and don’t need perches. While not 100% effective against Grackles, it will discourage them and make it hard for them to feed.