1997 Honda Accord Water is leaking into my car's motor

Engine Cooling problem
1997 Honda Accord 4 cyl Automatic 175000 miles

I was told awhile ago that there was a small leak in my radiator and that eventually I would need a new one. Last month, I was having to put coolant in my car every week so I finally took it in and got a new radiator. Coolant was still leaking from my radiator and now there is a milky discharge in the cap of my motor. I was told by my mechanic that the hoses were good, I also just had the water pump replaced, and the radiator is new and not leaking. As long as there is water in the reservoir, then the coolant level stays the same in my radiator, but it drops when water is not available in the reservoir. I was told by my mechanic that the only way to fix it is to send the head gasket to a machine shop and basically rebuild the motor. Is this correct? They told me it would cost $1300. Is this too expensive? My main question is: Will doing that fix the problem and is this the only thing that could be causing this problem? Should I just get a newer car? Thanks for your help.