I am assuming that they are welded in the up position - unless the car is never driven at night? If somebody got really creative with welding stuff up I can believe that the dealership will not want to touch them if they are seriously buggered up. There is no repair procedure in the service manual for fixing 'welded in place headlights'.
The 'hinges' are located at the back of the headlights. In addition to the hinge (really a pair of pivots) you have the retractor mechanism which drives the headlights up and down. You could put a tack weld on the pivots to try and hold the headlights up; but, they will be rather floppy unless something is done with the retractor mechanism to hold them in place. Do you still have the retractors mechanisms and are they operable or were they also 'fixed'?
If the pivots at the back of the headlights were just tacked in place it might be best to try and grind the tack welds and restore the operation of the pivots. This might require fabricating a new pivot pin. If the pivots are truly welded up solid you have a problem. The pivot assembly does not appear to be included in the headlight assembly. I don't know where in the parts catalogue it is located. You will have to do some serious searching in the on-line catalogues to find them and see if replacements are still available
https://www.acuraoemparts.com/v-200...ual/electrical-exhaust-heater-fuel--headlight
If your retractor mechanisms have been replaced with something solid or have been otherwise rendered in-operable you have a double sized headache. They are no longer available in North America. As noted by hondasash you will probably need to source them from Amayama.com which may be your best source for most of these parts anyway.
Two suggestions:
1) Post pictures of your headlight assembly with the covers removed so that others can see what creative wonderfulness has been visited on your headlights and advise on possible courses of action.
2) About 10-15 years ago it was trendy to try and 'update' your 2001 and earlier NSX to look like a 2002+ NSX by replacing body parts including the headlight assemblies. There may be some pre 2002 headlight assemblies (and all associated parts) sitting in boxes in garages looking for a new home. If you headlights have been really messed up it may be cheaper to post a wanted to buy add on Prime to see if any used parts are available.