Hate to tell ya, but small diameter drills do wander. And once you have the hole started, any additional drilling will simply follow the hole. I need a bit more info to help. How much work and where are you in your process? Barrel in, lock inlet, etc,
Draw on the outside of your wood where your barrels bottom is from the muzzle to the breech end of the barrel, without regard between the two points. (You only drill a straight hole). Once you determine this line, measure down the web thickness; between 1/8 and 3/16 depending on you. (I put mine in at a skinny 1/8", but that takes practice and has no room for "new builder" error); and draw this new line. This is the top of your rod hole and the base of your rod channel up the forestock. Your rod should be 2/3's exposed and lay out the dimension (whether 5/16 or 3/8). Leave a little extra for shaping the upper forestock along the rods finished edge.
I cut my excess wood off the stocks bottom so its not in my way and doesn't restrict the drill.. Now flip the stock belly up and mark out the rod channel on the bottom of the stock and cut a little groove up the forestock. I use a rat tail 5/16 wood rasp to make short order of cutting the channel out. Lay your drill in and use a couple pieces of leather shoestrings to tie the drill down 4 inches back from the muzzle and about the same from the entry hole and drill an inch and withdraw. Repeat. Use a little soap for lubricant, or beeswax though that gets sticky if you get into trouble. The drills which are solid tend to wander. Tubing with the brad point bit silver soldered into the end works fabulous and doesn't wander unless you steer it wrong.
Let us know how far you are into this project. Theres a lot of experience floatin around in here. We can help!
Susie