Will the government simply take what I owe the EDD out of my refund this year and I get the remaining or do I get my full refund and pay the EDD myself?

If you're subject to a refund offset, the government will take whatever amount is delinquent out of your refund, and you'll get any excess amount. 

The best way you can find out if your refund will "offset" any relevant debt is to call the Bureau of the Fiscal Service's Treasury Offset Program.  

You'll find the number in the IRS' Topic Number 203 - Reduced Refund.