I've got the same press (a couple years newer than yours) with the same chuck. You'd actually need two spanner wrenches - one for the spindle and one for the threaded collar. You pull back the threaded collar down against the chuck and work the two wrenches against each other.
OR the wedges / c-clamp method. I've used both. Your chuck may need a lot of persuasion to pop off the spindle taper, so the wedges may be your best bet if you haven't already gotten the spanner wrenches.
Whatever you do, make sure you have a "landing pad" for the chuck to drop onto when it pops off. An old towel or some rags on a block of wood directly under the chuck should do just fine. You don't want it popping off and shooting itself, with some force, onto your cast iron drill press table. Bad things can happen.
Once you get the chuck off, it probably is best to pull it apart and clean it. It sounds like it just has decades of old grease and dirt gunking it up on the inside. It may free up with a liberal dose of wd40 or some light solvent, like lighter fluid, while working it open and closed, without taking it apart. Given the age and symptoms, though, a proper cleaning would probably be best. The prospect can be a bit mysterious / intimidating if you've never pulled one apart. It certainly was for me, until i quit putting it off and went ahead and opened one up. It's actually really straight forward. There are numerous youtube videos on how to do it. I also recommend going to the Jacobs chuck website and looking up their directions for pulling apart / replacing the jaw and split ring for their standard chucks. That will set you straight, and the youtube videos will demonstrate how simple it really is to do. I don't have a proper press, but i do have a good bench vise and a good woodworking vise that opens quite wide. In a pinch, i've pressed them apart and back together again with some judicious use of a mallet and a block of wood.
Best of luck with it. Yours really is a nice drill press. The ones with the ships wheel, like yours has, are a bit prized by the vintage tool crowd. They're not very common and have a high cool factor.
paulh