On my 1970 - 71 Chryslers, it can be removed with the dash in the car but my recollection is that you have to remove the glove box, which in your car is like the 1970 Chrysler model (not the 1971 plastic/cardboard one) and it is all cardboard and will likely tear when trying to remove it (it is made up of an upper and lower section). Also, you will then need to remove the center a/c outlet duct assuming you have that option in your New Yorker and that can be risky, as doing that usually cracks the plastic outlet, and you also have to remove the cluster area upper plates. Doing all this allows access to all the various tinnerman nuts from the studs on the dashpad. If you are careful and your car hasn't been exposed to a lot of heat over the years, you might be successful in doing all this without a problem.
Otherwise, just allowing the dash to rotate downward (pivoting on the two bolts holding the two lower dashpanel pivots to the kick panel area) would also allow access to most of the nuts, but in doing so you would then risk breaking the plastic studs on the lower spot coolers as well when you try to remove the ducts from them (in addition to the center one) but might be able to avoid taking out the glove box I believe and avoid removing the cluster area upper plastic plates. Removing the center speaker metal plate on top the dash has to also be done with this second approach and also the small hex head screws that hold the dash panel againt the firewall also have to be removed, and a flexible socket tool (available from Home Depot) helps at getting straight on those little buggers, and they are usually pretty tight. So either way, the Chrysler pads are a bit of a pain, but doable.
Replacing the dash pads on the Dodge/Plymouth models is easier I have found just leaving the dashpanel in place and removing the instrument cluster and the glove box, as most of the nuts are accessible then and the plastic a/c outlets are sturdier than the Chrysler outlets and don't tend to crack when trying to remove the ducts to the outlets.
So on your 1969 New Yorker, take you time and just power through it until you get it done, and when you get frustrated, go take a break and resume and don't give up until it is done.