Well, cooking off the oil in 1K miles tells me that there is excessive gap inbetween your rings and the cylinder walls. Combustion byproducts are finding their way down into the oilpan and contaminating the oil. Since you dont have good seal, oil is also finding it's way up into the chambers and, heck, probably the valve seals are starting to leak too.
Coolant is so pressurized that if you did have a leak, it would consume copious quanities of coolant in a short period of time. Burning oil also has a fierce odor not unlike burning coolant. I fogged my engine so I am quite familiar with that lovely scent. Pull a plug, it should be black and carbon fouled. If so:
To duct tape the problem without an engine disassembly, do this:
1. 20w-50 (might be a bit harder to start) regular non synthetic. This will not go past the rings. It's good heavy oil.
2. Add some of that "engine restorer" stuff in the silver cans. Makes a tiny bit of difference but I have always found it to help.
See if this greatly reduces the problem. If not, add some lucas oil additive. It makes the oil gooier still and it wont flow as easily past the rings but will tend to stick to them. Make a tiny difference too.
Good luck