The fact the system gets cold as the Freon is going it means next to nothing. Any system that is empty will get cold as it is being filled. Even if the compressor is not working because the fluid is in its compressed fluid form already. As it enter the system it becomes a gas and very cold. Until the system pressure is equal to the pressure in the storage can the system is getting colder.

Ok maybe this will help with the explanation. If I took a fence post and welded the ends close except one fitting, I then sucked the air out of it like a AC machine would do and I filled it full of Freon and you tried to hold it would freeze your fingers to it. In fact it would get colder then any functioning AC system would. It is not uncommon to have system ice up while being charges even before the system is turned on. So based on the info in your posted there is no way to give you direction.