Australia has experienced the same issue, much of our recycling was shipped to China. No more. How it was handled varies by local government area. My local council didn't have anything going to China, they used all domestic recycling facilities. The recycled bundles sent to China were constantly being devalued because of contamination (e.g the paper bundles would have too much plastic, or wax, or oil or other contaminants and this made much of the bundle useless for paper recycling as the extra level of processing wasn't worth it). The lower the "purity" of the substance being recycled, the less valuable it becomes.
In my state they have recently introduced the Tomra "reverse vending machines" - which accept cans and bottles (aluminium, glass and plastic). You feed the machines items one at a time, it scans it and either accepts or rejects it. You get 10 cents for each item accepted. This is working pretty well and sorts out very quickly what is and is not accepted. It provides for a low level of contaminants in the items collected making them sufficiently valuable.
You can choose to have the value of your returned items credited to any of a list of charitable organisations, or deposited directly into your PayPal account.