Hello & Welcome to JA,
This problem is usually the multifunction/turn signal switch that has an open circuit.
I must ask about this part of your post "selonoid buzzing like it could not hold the position of the headlights being on"
This is an S-10,right. There are no headlamp doors,right ?
The headlamp circuit does not have a solenoid.
This is a 1996 Chevrolet Blazer S10 4 door 4x4. It is a first model for that year. The first model came with Glass headlights and the second model for that year came with the composite headlight setup. I have the first model which has the glass headlights. Also it is the model that has the manual 4x4 shifter mounted on the floor not the push button type. The headlights on this model have day time running lights. There is a light on the dash shaped like the headlight that lights up green when the running lights are on. This light goes out when you switch on the headlights. There is a solenoid that switches the running lights off the headlights on. I know this because before I replaced the headlight switch I could hear it buzzing, the light on the dash indicating that the running lights are on would flicker to this buzzing. The solenoid is located in the glove compartment. I tried replacing that solenoid also. There are three of them in a row in the glove compartment. I purchased one new one and rotated all of them and could not find a bad one. One of the solenoids controls engine management because when I unplugged it the truck shut off. I am going to have to chase the headlight supply circuit to find where it is supplied from and then chase that up to the headlight switch and look for any breaks. It cannot be the ground side because the high beams still light up when you turn them on with the multiswitch.
Here a couple of schematics that may help you.
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