Diagnosis
The Ash Drawer is binding due to thermal expansion or a warped felt seal. The Classic III's drawer fits inside a metal frame that can distort slightly with heat.
How to Fix It
Ensure the grill is cold. Do not force it. Tap the front face of the drawer firmly with the heel of your hand. Simultaneously, wiggle the drawer side-to-side while pulling straight out. If it moves even a millimeter, work it out slowly. Once removed, inspect the metal frame for any obvious bends.
If That Didn't Work
Remove the three screws on the inside front of the firebox that secure the drawer frame. This allows you to remove the entire frame-and-drawer assembly as one unit. You can then free the drawer from the frame more easily for inspection.
Heads Up
Do not modify or remove the felt gasket on the drawer. It is essential for creating a proper seal for the air intake system. A missing gasket will cause major temperature control issues.