Press stops under even slight pressure on drill

if you are using a normal 3 jaw chuck on the drill , that is normal when hammering
the vibrations loosen of the chuck
a method to stop it is to ensure the chuck is tight and keep pulling the drill out of the hole to remove the grit that jams the bit

There are several possibilities. I am assuming that by "hang up" you mean that the motor still runs, but the arbor, chuck, and bit all stall. I also assume that you are running the bit up and down to clear out the wood shavings from the bit out of the hole, you are using the correct amount of downward force, and that the bit is sharp. I would first check to see that there is proper tension on the drive belt. You can add a belt spray designed to increase friction for better running. I would also gear down the drill press even more to increase torque (I know you say you are running at the correct speed, but try a reduced speed). Another method is to use an auger bit or a forstner bit without pilot holes. If you are using a spade bit, you might try no pilot hole, and drill at a faster speed. I have a junk 10" drill press and have no trouble drilling 1" holes in oak.

The chuck on 99% of the drill presses is just a press fit. The end of the spindle where the chuck goes on is slightly tapered to hold it on. Make sure the end of the spindle and inside surface of the chuck are very clean. With the motor off, raise the table high enough so you can still get the chuck on the spindle then turn the handle to lower the chuck against the table and press it firmly against the table to wedge the spindle in the chuck. There shouldn't be any need for anything more than this.

that depends on whether or not you have a keyed or keyless chuck. if you have a keyless chuck then you do not need another tool you must turn the chuck counter clockwise to loosen the chuck and replace the bit. if you have a keyed chuck then you will see three holes around the chuck of the drill ie the part that holds the bit. just below the holes you will see teeth spaced evenly around the drill. a chuck key allows you to put the end of the key in one of the holes and using the teeth you turn the chuck key thereby loosening the chuck and allowing you to change the bit. if you have a keyed chuck just take it to your local hardware store and they can provide you with the correct key and show you how to remove the bit. if your chuck is keyless the hold the chuck put the drill in reverse and pull the trigger use a glove to protect your hand

You have to tighten the chuck more. Tighten in all three holes to get even pressure on the bit.

check the screw in the center of the chuck it could have come loose causing it not to lock properly.

You may try to put a thick soft metal pipe (aluminium or copper) over the drill bit (longer than the bit itself by ~10 mm) so it will touch the chuck's jaws, then GENTLY hit the other end of the pipe with a hammer several times, holding the pipe a bit askew, thus trying to disengage the jaws from the lock in their thread, and try to loosen the chuck after each series of hits. Just be careful with your fingers, and better hold the drill by the chuck while hammering it - so the hits won't go to the transmission unit (sorry for description - hope it's understandable, if not - ask for clafication from me)

When you buy a replacement chuck it will have detailed instructions- at least Jacob's brand chucks do. Basically there is a retaining screw inside the chuck that needs to come out first- it may be left-hand threads . Then you put the short end of a 1/4" or larger allen wrench in the chuck and close it tightly. Set the drill on a bench with the chuck off of the bench and strike the allen wrench with a mallet in a counterclockwise direction. Do this several times, then you'll be able to spin the chuck off by hand. Reverse the process to install the new chuck.

There is a long tapered hole in the spindle of the press known as a Morse taper. It holds a 6 or 8 inch morse taper arbor which is attached to the drill chuck. This is made to be removed frequently by putting a wedge shaped key into a horizontal hole in the drill press and giving it a little tap. This is for making tool change that involve removing the chuck. If you had two such tools (say two different style chucks) they would each have their own dedicated morse taper arbor. But there is also a short tapered cyclinder on the chuck end of the morse taper arbor, with a matching hole in the back of th chuck. This is not intended to come apart frequently, but it can be disassembled. I'm not sure which one is coming apart, but regardless of which one it is, you might try this fix. Clean up any bumps at all on the oustide of the taper and inside the hole with sandpaper 200 - 300 grit. Use a dowel to hold the sandpaper in the hole, and use the motor to spin it. --- JUST BE CAREFUL that if it grabs that it doesn't grab you! Once it's all cleaned up, you can you the press aspect of the drill press to seat it in place. You also might have to replace the arbor. They don't cost all that much and are available from Grizzly.com. You specify the particular taper for BOTH ends of the arbor. The long end is a Morse 2,3 or 4, and the short end is specific to the drill chuck manufacturer, like JT-2 (Jacobs chuck taper 2). Write back if you need more help.

Something I would try:

George, look down the center of the chuck jaws with a bright light. It's going to be difficult to see, but there is an Allen head screw down in there.
Taking the screw out, will afford you the opportunity to remove the chuck from the drill. The drill should have a slightly tapered shaft, and the chuck has a tapered hole to match. It might be stuck on tight. Tap on the back side of the chuck, with a wooden block, and hammer.
(Yes, this may require a helper)

Tap on one side, then the other to try to remove the chuck evenly.
With the chuck off it may release pressure on the jaws. It may not.
In this case the chuck will have to be opened up, (Which will release pressure on the jaws), and the jaws reset.