dtwbcs,
I would do a compression check on it after it's real warm. Some overheating could be associated with timing but yours almost sounds like a blown head gasket.
A functioning radiator should have the capacity to cool it down even after getting hot, could yours be plugged or dirty.
When a engine get real hot, all kinds of things start to happen. Marginal ignitions fall apart, fuel systems start to boil and act strange and things just get weird. If you are using coolant, this could support a blown head gasket or cracked head theory.
Stuart