Whether to take it to a shop depends really on your own comfort with doing the work. The problem can certainly be fixed, and the fix is actually comparatively simple, but you want to do it with a drill press and wood plugs, not just dowels. You want to use plugs because the screws grip the grain better when they run perpendicular to the grain. You don't get that relative orientation of the screw and grain with a dowel.
What you would do is bore out the hole to a diameter that will accommodate a plug cut from across the grain of a piece of lumber thick enough to fill the hole. Coat the plug with Titebond glue and jam it in the bored-out hole. You'll scrape the plug down flush with the heel of the neck, and then re-drill the screw pilot holes where they're SUPPOSED to have been. Alternatively, you could install threaded inserts and use machine screws, but that's an additional bit of work.
This is probably the tool you'll want to cut the plugs out of your piece of scrap lumber:
http://www.woodcraft.com/product/146724/plug-cutter-14.aspx
As always, if you lack the skills to do this on the first tray, practice on scrap. Make sure your drill press table is level and perpendicular to the bit, and make sure the neck is clamped in so it, too, is perpendicular to the bit.
