I know I don't own an Accord, but I had this very same problem with my Del Sol. I can tell you the fix for the Del Sol. If it works, yay; if it doesn't, well I tried.

In the Del Sol, RIGHT under the glove compartment, there is a plastic piece. Coming out of the piece there is a rubber tube that goes out the firewall. Reach your hand under your glove compartment and dash and feel around for a round rubber tube. Squeeze that tube and examine it for any pinches or cracks. Mine was pinched. I squeezed it and heard this bubling noise of the water draining out, and that was that. I never had a single problem since. Check it out under that dash of yours. It may be similar. I know I don't have an Accord, but you never know.