We've had a medical bill go to collections. I can't afford the amount of payment the agency is requesting, but I am making what I guaranteed and a bit more when I can. Today we received a notice that if it wasn't paid in full in 10 days of the notice it would go to court for garnishment. Can they do that if I am making payments as promised and they are cashing my checks?

You are only making the payments you said you would make, not an agreement that they have accepted and that agreement is not in writing.
So if this is not a agreement that both you and them have agreed to, yes they can go and sue you for a judgement.