Firstly, I think you may be trying to use metal twist drills on wood, which as Harry says is not going to work all that well. When drilling wood the drill bit needs to have a "nicker" on the outside of the circle you are cutting to shear the fibres before they are lifted. You also need some type of centre point to keep the drill bit going in a straight line. The modern (i.e. post-WWII) tool to use is called a "brad point twist bit", also known as a "lip and spur bit" which is similar to a metal twist drill, but is modified at the tip:
Above: Lip and spur or brad point twist bit for woodwork
Below: Jobber twist drill for metal
The right type of drill bit is fortunately cheap - take a look at
Tool Station's offerings, but they normally only go up from 3mm to 10 or 12mm
I think the type of bench drill you are using wasn't really designed for wood, either, or at least not in the way we think these days. Most of the old wood bits you'll come across have a 4-sided (square) tang at the end and are designed to be used in a brace:
which generally require a 2-jaw chuck with a tapered receiver on the inside of the shell to hold and drive them. Unless you saw off the tang they won't work in a metalworking 3- or 4-jaw chuck. The other issue with woodworking centre and auger bits is that they invariably gave a threaded point, which is designed to pull the bit into the wood:
(Note the spur which slices the fibres of the timber).
Your drill is designed to self-advance as the handle is turned so any drill bit with a guide screw will be fighting the feed mechanism of the drill.
Given that your type of drill was on sale until the 1950s, what were they used for and what type of drill bits were they used with? From personal observation I know that they were at one time used in printers and print finishers for drilling stacks of paper and card, as well as for drilling lead (type) and zinc plate (etched printing plates) so that they could be attached to wooden backer boards* before being mounted on a platen press or even on the platen of a screw press. I have also seen them use for drilling holes in horn (think shoe horns which were at one time made from bone). I can therefore imagine that with the appropriate bits they would be able to tackle a lot of different drilling tasks, but not with most (all?) drill bits you'd recognise today, which are designed for power tool use. Some of the bits used (at least for paper and card) resemble spoon bits with the ends ground off, others are like the drill bits once used as pilot bits in "Yankee" push drills:
and there are also D-bits (literally a piece of drill rod cut off at an angle and ground away at the end to give a cross section which looks like a capital "D" when viewed end on)
* - with aapologies to anyone who is a printer - I am not and I obviously don't know the right terminology