Here's my similar experience (From a previous post):

This just happened to me also (1991 Accord DX, 153k miles). I noticed
that the car is fine while the key stays in the "Start" position (the
engine catches), but the engine is killed when the key returns to "On"
position, which is the norm for driving. To make a long story short,
I changed the Main Relay to no avail, but have since tracked down and
replaced the "Steering Switch" (35130-SM4-305 for my model). This
switch is essentially the ignition switch that sits on the left side
of the steering column. Replacement work is straight forward and took
about 20 minutes if you have the part in hand. Simply remove the top
and bottom covers off of the steering column (5 screws total), and
find the white pastic switch on the left (It almost looks like a
bearing housing). Replace this and your car should work like a charm
again.


I live in the Bay Area, California, so purchasing from San Leandro
Honda (slhondaparts.com) was a cinch. I ordered online and picked up
in store, therwise you'll need to pay for shipping and taxes. The
cost of the part is $50, and I got a similar quote from hparts.com,
when they were still in business.