What would cause the right blinker on the dash stay on but only when the headlights are on and blinker still blinks ?

Blinker-relay is malfunctioning. Its mostly located under the dashboard

It is hard to say for certain without knowing the year, make, and model of the vehicle, but I would look for wiring problems in the steering column and blinker switch, especially if the headlights and blinkers are both controlled from that location.

shorted wiring usually under steering wheel. .

That would be caused by a bad ground connection to either the front or rear parking/turn (probably rear) signal bulb socket. When the ground is bad, the parking light filament with find a ground through the blinker filament, then through the turn signal indicator, keeping it turned on. I have seen this on many cars over the years. Hope this helps, good luck.

Plain and simple...Your turn signal circuit is shorted to your headlamp circuit. What USUALLY causes this is an incorrect light bulb has been installed in the vehicle or the light bulb was installed incorrectly. There are other possible causes of these circuits being shorted together, but 99 out of 100 of the times that I have seen this it is either because somebody just replaced a bulb or a bulb socket melted somwhere on the car. Be sure to check both the front AND rear bulbs.

It sure sounds like you have a case of "sneak circuits"
These are usually caused by a bad ground somewhere and something is "grounding" through the light bulbs.
Have you changed any wiring or installed a stereo or some other accessory?
You might use a jumper wire and ground the dash and other areas to the negative battery post as a way to find the culprit.

make sure our lights at rear are grounded sounds like a bad earth

check the relay and blown out bulbs will cause the lights to blink fast or some to stay on. one bad bulb will cause the problems.

you have a left side turn signal out in the front or back replace bulb the reason why your bilnkers wont blink is because your flasher relay is bad the relay will be located n the fuse panel under the drivers dash any auto parts store should have the light and the relay in stock replace both of those and youll be in buisness

I have had this happen and it is rare, but I have had a filiment burn out in a two filment buld and short to the other filiment. It does happen and I think this might be your problem, the only way to find it is a visual inspection of each bulb. I am sure many people will think this is crazy, but I have had the exact same problem, I had the schematics, tested the circuits,etc. and them by accident found a stop/blinker 1157 bulb that had done this exact thing. I hope this helps you, check the bulbs in your headlights also.