As for the wobble running the drill in a electric drill, I run 300mm long 3mm diameter drills in a milling machine at 6,000 rpm and don't have a problem. Either the drill is bent or the drill chuck is not running true or is damaged.
I think a 300mm bit in a hand drill drilling at a near horizontal angle might be a different story than vertically in a milling machine.

I only use a battery electric drill to drill holes but to reduce the wobble dont start the drill unless the end is against the wood, even if you use a normal length drill by hand and twist it against the wood to give the drill a place to start there shouldn't be a problem. Just dont try to push it through the wood too hard and bend the drill.
I am also talking about a cordless electric drill, and know well enough not to start the drill before contacting the material being drilled as well as letting the bit do the work. Pretty much rule one with most any hand tool.