When we moved into our house, the furnace was filthy. Previous tenants didn't use filters. While removing a grocery sack full of dust, dirt & lint, the balancing weights on the squirrel cage came off. They didn't appear to be in great condition either. Where would I take the cage to have it balanced? Is this something I can do myself?

Cages are not rebalanced they are replaced. The cost to rebalance would be more than a new one. Contact your local HVAC company for a replacement or if you have the exact model and brand of unit you can search on Google to see who has parts for your unit.

And get you ducts cleaned after it gets fixed.