Drilling is always VERY dicey for me. Even with my little drill press, and anyway very often I'll have to use the hand drill because it won't fit in the thing or whatever.
....sometimes I can't see around a larger bit so it's off when I'm done... walking bits... and depth is hard to control too, like when you just want 1/16 like this pic.
I suck at drilling.
Then most of the time I have to go with it after the damage is done as you may know.
Thanks Richey, marking up two fret boards later!
I'd agree with Ritchie for the most part.
Using general purpose drill bits for fine work is going to give you unnecessary trouble.
Invest in a small set of brad point bits. I deal with Rockler.com and Woodworkersupply.com for stuff like this. The local hardware/big box store rarely has such things.
Other than that it looks like you are marking carefully. Make a small pip on target with any sharp point, and the tip of a brad point will be guided by that. Use a variable speed drill and start very slow. When the bit begins to cut at full diameter, begin to just "blip" the drill a moment and check depth. I assume you are leaving your dots a shade "proud" and sanding/polishing them flush.
With a little practice on some scrap you should be doing ok pretty quickly.
FWIW, I have found most small inexpensive drill presses to be too sloppy and imprecise to be of much help for a lot of this kind of work. Used correctly and carefully, a hand drill can be used for many such things.
I'm a big user of a sliding mitre square as i have worked in metal fabrication for years. usually mark with an awl too, i do like a bit of human error tho. dont like my builds too clinical.
Wood bits on hand drill make cleaner holes if you use the drill in reverse.
Get the center by using a tiny screwdriver and don't drill in but lean on the bit and drill backwards, the tiny cutting ridges leave perfect holes and slice into the ground of the hole without ripping into it. Try it. It works. Not sure what it will do if you use the same technique with a power drill. But do it very very slowly.