1996 Dodge Neon Cracked Block? Gummed up Radiator

I've been having an oil leak for about 2 years now. I havnt fixed it because its a neon, I'm not willing to put money in a $2,000 new car. Last christmas the car started to over heat. I opened the radiator and saw yellowish brown mixture in my radiator that feels a lot like oil. I imediatly flushed the system and refiled everything. It ran fine for a year. I did notice that over time the coolant became darker and darker, turning black, then eventualy the brown yellow color. I flushed the system again. A month later it started over heating again, I took out the thermostate and it still overheats. I see the hoses bulge where the coolant mixture is backing up, its so gummy it cant make it to the engine. I took the car to a repair shop to see if they could see whats wrong. They said that my radiator is so ate up that there is a leak from my transmision into the radiator. I have checked the transmition and it has not lost much if any fluid, but my oil is always disapearing. I do see spots of oil on the ground. They say if I change my radiator it will fix the over heating problem, but it doesnt make sense to me. I have not had antifreeze in my oil or in my transmition. What could make my antifreeze turn this color? Most people I ask say a cracked head, if this is the only other cause than RIP. Is it possible that this could be the problem for the overheating, and the oil leak is an entirly other problem?