Before I answer, I need to be sure what you said and additional questions.
1. When you step on the brake pedal, do the correct filaments on the brake lights come on?
2. You say the dash indicator light comes on. Does it flash or just stay on when brake pedal is depressed?
3. Do all fright and left turn signal lights flash as they should? Does the dash indicator flash as it should? Try this with brake on and with brake off.
4. Check as per glenns towing suggestions.
5. Report results/answers.
Wiggle the turn signal lever during the problem you mention and see if anything changes. Check the condition of the brake light bulbs and their sockets.
The brake lights, the turn signal lights, flasher and dash indicator are all routed and controlled thru the turn signal switch.
On a truck that supposedly has had no work done either in the past or recent, the common issues are ground (hard to find), bad bulb, corrosion, loose wires, turn signal switch.
Corrosion can be in the ground paths or the power paths. It can be located at screwed connections, solder connections, fuses, bulb sockets and connector contacts.
The answers to my questions are a start at troubleshooting and will result in the next steps from me and others. You will need, at a minimum, an ice pick style circuit checker.
Sometimes we tell posters to "check the grounds". That is much easier to say than do. There are dozens of places and many are hidden. Lets get a few answers first. It is true that more than 50% of the time it is a ground problem because there are almost no wires in the ground path. It's all "intimate contact".
