I'm thinking your ignition switch connector is bad. That is a common
symptom, I had the same on my 1987 740 GLE.

http://fcpgroton.com/volvo700900steering.htm

9447805
Ignition Switch Electrical Portion fits: all 740 Models 85-92; all 760
Models 83-87; 780 V6 1987 only; 780 Turbo 89-91; all 940 91-95 Models
excluding S.E Models

$62.00

Remove the panel under the steering wheel. Unscrew the radio amplifier
mounted under there and just let it hang out of the way.

Lie comfortably on your back using a picnic table bench, or long ice
chest and have a flashlight handy. It is simple to remove and install
this piece, it can be tricky to get the screwdriver in there and on the
screws. I use a straight flat blade screwdriver and also the
screwdriver tip and adaptor for a hex-head screwdriver.

Unscrew the 2 screws on the ignition switch  and pull the device
straight back. Re-install the new device the same way.

G'Luck

Jamie

Genuine Volvo

You can test this by turning on the ignition and turn the key slightly
back, not enough to  kill the engine. Also wiggle it a bit to see if
you get a connection.

I'm 99% sure this is your problem.