Have you tried drilling 90 degree holes? You might find it comes naturally or with practice.
You can make a simple drill guide out of two pieces of wood glued together at 90 degrees to each other and positioning the bit in the corner between them. Alternatively if you get one good hole going through a piece of wood at 90 degrees, that can also be used as a guide. It wouldn’t last for ever as a guide, but it wouldn’t have to given the quantity of screws you’re talking about.
If you are planning to unscrew the pieces many times, you might consider using threaded inserts and countersunk head socket screws.
