You have a leak in the transmission cooler that is inside your radiator. And yes it's very possible it has ruined the clutches in your transmission. You definitely need to get the radiator replaced and the transmission flushed multiple times. Then cross your fingers and hope that the transmission survived.
bottom line get the radiator fixed right now.
.
2001 Suburban 4x4. 6.0L, 4.10 3/4 ton **** 2005 Jayco Jay Flight 27BH **** 1986 Coleman Columbia Popup