Is the water greasy feeling? Did you mention this because you suspect a cooler leak?
Daughters 96 Jeep (318) had a trans cooler leak into the radiator and the rad fluid had a brownish red color but was greasy feeling and stained the overflow bottle red.
I agree with others, Drain, refill and see what you get after,
Is there signs of water in oil?
Regarding heat (others chime in here Please ) isn't that sort of normal during break in?
