Hey try checking you radiator plugs (2) at the bottom right corner make sure its not stripped. Also if the radiator in plastic on the top and very bottom but iron or aluminum in the middle you will have to check really careful where the plastic and aluminum join together could hair a hairline crack but once you start the car and it get hot and pressurize then its runs out fast. Hope this helps I have a 93 accord also..for 12 years now and still doing well. This was the solution offered to me by one of the experts in Fixya. Here's the link to the site:
1993 Honda Accord V6 Sedan - Leaks fluids - FixYa.
I hope this helps!