according to my cartridge conversion book, 38 Long Centerfire cases can easily be made from either 357 mag or 38 Spl cases. trim to a case length of 1.03 inches and size in a 38 Spl die. a 38Spl case length is ever so slightly longer and is 1.155 and a 38Spl may fit the chamber without trimming if the chamber was cut with even a little extra room. The empty case will fit because the chamber is straight, with the bullet added you might be pushing the bullet into the rifling to close the breech. The difference is only a tenth of an inch.
As for bullets, you will need a heeled 375 bullet. The alternative is breech seating the bullet into the rifling and just loading a case full of powder behind it. (be careful to leave no gaps between the powder and bullet.)
Folks have tinkered for years about ways to reload the outside lubricated bullets. I suppose a small gizmo could be made to shave a lip around the base of the bullet. The old Ideal nutcracker tools for 38 long centerfire do come up on the big on-line auction perhaps once or twice a year. They have a mold incorporated in the tool. I needed one in 32 long centerfire and got one for $35.00. The alternative would be to have the chamber recut for a cartridge that is availible and takes a modern style inside lubed bullet, like the 38-55 Winchester. There are a few other 375 bullet cartridges that would fit the bill, but would require making cases from others, 38-50 Maynard, and 38-50 Remington.
Is it a first or second model Whitney? If it is the first model, the action is not strong enough for many modern cartridges even when loaded with black powder. Frankly, I wouldn't shoot a first model in 45-70 or 40-65. 38-55 with reduced loads would be max for the first model.