I'll give you two ways to do it...
I prefer the cut the ends off on a spinning lathe, thus leaving a small centered nub, by cutting off the sqare ends now I save alot of time but maintain my center for referance pourposes...
first is on a drill press, depending on the heigth of your drill press, you may or may not need the spacer blocks I show in the photos...
first make a board to fit your table on the drill press, directly in the center, drill a 1/4" hole and install a 1/4 dowel center, with the table low enough to acomodate a the plug and a 6" drill bit, use either a long drill bit or a 1/8" steel rod that has been ground to a point on one end, line the drill up to the dowel center by loostening the bolts on your wooden platform until the rod and the dowel center meet, lock up the table...
put a 3" drill bit in your drill, set the speed for approx 1200 for soft woods, 800 for hard, line up one end of your plug on the dowel center and drill from the opposite end, switch ends and repeat... now , put a 6" bit into the drill and repeat, be sure to clear chips as you go or the drilled chips will deflect the bit, use a slow advance , if needed the blocks, each of which has a 1/4" dowel in the bottom and a hole drilled for the dowel center can be used to "raise" the table without loosing your adjustments, on drill presses with a high table and short stroke, you may need to run the drill into the plug, then with the plug still on the bit, raise the drill head and insert a block...or you can finish the job with a long hand drill...
Next is the eaisest way, using a drill chuck in your lathe,install a 6" bit, lathe should be set to 1200 for soft woods, 800 for hard woods.. line up your drill bit plu and tail stock, the lock lever should be adjusted so the tail stock slides eaisily, switch on the lathe and SLOWLY advance the plug forward, reversing direction often to clear chips, once the end has been reached, switch ends and finish the job....
For the belly holes, I have a jig that lines up the lip slot or, if reversed, has a screw that holds the plug in place via the through hole, a removable tail stock accomodates any sized plug, by have ia adjustable fence on my drill press, I am able to slide the jig in place against the fence, thus the plug is centered under the drill for each operation, with no guessing or adjusting needed, spotfacing, and drilling hanger and hook holes takes less than 1.5 minutes from the time I grab the plug until I grab the next one...