There is only one place to send it for a CLA: Sover Wong. He did mine, and I'm very pleased.
I'll let others recommend a grease, but you may just be able to clean the gunk off and get it working OK.
The other thing to check for is deteriorating foam seals on the top and bottom back cover slots and the mirror damper cushion foam. Sover will replace those, but if you want to do it yourself, Jon Goodman sells an excellent kit for $10 which, with some care, you can install yourself.