On my 62 Starfire, I just installed the front grill and headlights. Low beam, both outside light are fine. Hit high beam switch, kills low beam and high beam does not light on the right side. Left side is OK. Head light switch and headlights are new, both ground wires on either side are connected to chassis sheetmetal.

 

First of all, the fact that at least one high beam comes on says that the dimmer switch is good. Be aware that when the high beams are on, you should have all four headlights illuminated - the low beam bulbs have two filaments in each and the high beam filament in the low beam bulb should come on at the same time as the high beam bulbs come on.

Similarly, if both low beams work, your grounds are good. Have you verified that the high beam filaments in the three bulbs that don't work are, in fact, good? If so, then check the contacts in the headlight connectors. Chances are they are corroded.

 

I've you've got a multimeter or even a test light, check that there's power to the connectors at the headlight bulb.
I wouldn't be surprised if the new bulb is defective.

 

I have low beam (outside light) on both sides. High beam switch turns off both headlights on the right side, not just the high beam headlight. Both headlights work correctly on right side.

 

Remove the connectors to the bulbs and verify a good contacts are in each plug. They shoul not be burned, pitted, or bent. If the connector comes off easily, then that could indicate a bad connection.
Check the bulbs themselves with an ohmmeter - new ones could be bad even.

 

The lamp sockets have new connectors and they fit really tight to the new T3 lamps. I will check for voltage on all four lamps.
Thanks.

 

Since you have all new connectors, compare them to each other make sure the right colors are in the right holes.
They should be right, but you never know...