Every once in a while I'll lace a wheel and it looks like the spokes may be way too long. Invariably it is the result of me not paying attention when lacing and getting one spoke in the wrong rim hole, which then affects the rest. Easy enough to fix though. I'd recommend that you take a good look at your lacing before you do anything else.
I agree with the recommendations to tension the wheel and see where you get to. Spoke ends protruding through the nipple can be managed by grinding if too high. The bigger issue is that you run out of threads before reaching proper tension, but you won't know that until you try. I, too, have invested in the Hozan threader and have found it useful for running a few more threads on a spoke. I have also cut long spokes and rolled new threads for a new wheel build, but that's a real chore to avoid if you can. I'm damned by being cheap and having too many spokes left over from prior builds.
Good luck