I have a 98 camaro and it has the same problem. I found that if i took off the top cover for the shifter i can push down on the housing and my high beams will come on.. I found a small plastic piece that came off. Try opening and looking. I heard that alot of camaros have this problem. The switch goes bad.
The 96 Camaro's high beams wires do not run anywhere near the shift lever that is on the floor of the 1996 Camaro. It is built into the turn signal selector which is on the steering column. Your problem is either your relay (under the hood in the little box on the driver's side, you have to remove the top to access it) or your turn signal selector. Does the turn signal selector click when you pull it back? If it does click you probably need a new relay. If it does not click you will need to pull the steering wheel to get to that selector and change it. Also look behind the turn signal selector there is a small wire connector that if it comes loose your high beams and cruise control will not work. Often you can connect these wires and get everything back.
http://camaroforums.com/forum/93-02-general-41/high-beam-issues-58085/
hi-beam switch is faulty
Check the dimmer switch and the electrical connections.
Defective light switch.
Check bulbs Could be bad switch (hi-lo)
try checking ur high-bea fuses and relays
check the wiring at the hi-low switch.
Normally caused by faulty dimmer switch.
steering wheel has to be pulled off and then you can take the top cover off the steering column to get to the screws that hold the switch in.
Your headlights turn off when you turn the high beams on because you have a defective switch. You need to have a mechanic replace the high beams switch. It is a simple replacement.
If the high beam switch is broken that it should be replaced. When the unit is replaced the high beams will work again.
I like to switch around 200-300 feet away from them
Your high beams work but your low beams do not because the switch is broken that switches between high and low beams. Electrical switches wear out.