It sounds to me that you have two things going here. First, I'll bet the drill bits you're using are "jobber" length. These are the most common and they are fairly long. Because of their length they tend to "flex". The simplest way around this is first: Insert them ALL the way in the chuck to reduce their length and second: Get yourself a set of these
http://tinyurl.com/6ccotu
I use center drills EVERY time I drill on the lathe.
After you drill a center hole, if you're drilling a hole any larger than 1/4", do it in stages. For a 1/2" hole I use 3/16", 5/16" or 3/8" and then 1/2".
I also highly recommend "screw machine" length drill bits. These are about 1/3 shorter than jobber length and I use them almost exclusively on the lathe. Cut your blank in two before you drill and you won't need a long bit.
I also use these (can you tell I don't like off center holes?:biggrin
http://tinyurl.com/6lp2ut
I bought these for drilling casings but now I use them on everything. A set of four, 1/4", 5/16", 3/8" and 1/2" will serve you well for making pens and you can get all four for $20.
At the very least try some center drills before you go spending money on new lathe parts.:wink:
Also, 1500-2000 rpm's is WAY too fast. 500-750 rpm's is where you want to be.
I also agree that you should tighten your tailstock lock slightly. Not so tight that it's difficult to move but you'll be able to tell when you have a slight drag on it and the slop is removed.