I also have an S10 96 Chevy Blazer and had the same problem. You need to replace the "Combination Switch" located in the steering column. It cost me $485 just for the switch. I don't know what state you are located in but wherever you are you might get it cheaper.
Check the brake light switch.
If your signal lights still work, the multi purpuse switch is probably bad, (turn signal stalk)
It might be that something has gone wrong with the physical power lock button in the car - the remote will continue to operate the power lock in that case. (Also, please accept my condolences, as a former Blazer owner, for having to be involved with a Blazer.)
Assuming the signal lights still work, no. But a bad brake light switch or a blown brake light fuse could.
Check your starter, and all power connections.
There is a short in the wires in the turn signal circuit. The combination switch in the steering column is a good place to check. Do the hazard lights still flash?
Usually this a faulty signal light switch. The Signal light switch disconects the brake light on the signaling side and makes it flash.
Depending on what type of vehicle it is, it could be the combination flasher relay. 2000 Dodge Caravan, same problem. Replaced DRL/Flasher Relay, problem fixed. Was an expensive dealer only relay...
A wire may be cut from the brakes. the turn signal wire is still connected
If the fuses and bulbs are okay I would blame the signal switch assembly. They call it a multi-function switch these days.
I have a 1998 Chevy Blazer with the same problem I traced the stop light wires back to the turn signal assembly. By taking the steering column plastic off you use a torx bit and a pair of vise grips to get the bolts out of the turn signal assembly. Otherwise GM says you need to pull the steering wheel . They still recommend you pull the airbag fuse. I pulled the fuse and disconnected the battery. I found a slightly melted connector behind wiring connection point. With a replacement turn signal assembly the brake lights now work. Be sure to find a aftermarket or salvage yard switch due to GM wanting alot of money for OEM.
Problem may be in the steering column (turn signal stalk)