This may be a little advanced for you but it is not that difficult. Many people drill the holes on the lathe. Do you have a 4 jaw chuck? And adding a drill chuck to the tail stock and you will be able to do what it will take a $150 - $200+ drill press to do.I have plenty of “practice” wood.. I just need to get a drill press, or something.
You don't really say the reason(s) you are having troubles, there can be so many factors/reasons, each one has to be dealt with separately and pics do really help a lot.
I have that fear if I think of asking SWMBO for another tool.I had to overcome the concern the lathe would grab my gouge out of my hand and beat me with it.