You need to cut at low speeds..
Yes.. copper is gummy.. I've done enough drilling and tapping on it.. Lots of coolant or oil and keep it slow.
I am no machining expert by any stretch.. But the problems your having seem to be the same mine were..
Keep the copper cool.. Not sure but I would bet the colder the better..
Keep the tool slow and make sure chips have a good exit. I would actually clean my drill after each hole because there would be a small build up of copper around the cutting edges. Probably a result of not being able to remove enough heat?
I think your on the right track..