You have 2 options: pay them ( upfront ) to take your furniture or find a mover yourself.
In both cases, there is no guarantee they take your items.
I'm lucky to have a cargo van.
Kitchen stuff was packed neatly. Big mistake. Tax receipt was for 10$ for 11 pots and pans, as everything could fit into one box.
Bottom line: if moving cost exceeds 20% of that charity donation, don't bother.
You'll lose money.