One more note about screws: The best screws to use are Sheet Metal Screws, rather than Wood Screws. They are available in the same sizes; that is, you can get them in #8 x 1" flat head Phillips. The advantage is that Sheet Metal Screws are made from steel and are hardened, because they are made to be driven through sheet metal. Most Wood Screws, like the ones that you bought, are just unhardened mild steel, probably 1018.
The Sheet Metal Screws are much stronger, less likely to snap off, and less likely to strip out the heads. Also, their threads are sharper and a narrower "v", so they will dig deeper into the wood without as much pressure to split it. I use Sheet Metal Screws for everything on instruments where I'm fastening things to wood: Neck screws, strap button screws, pickguard screws, and especially the teeny little tuner screws. I recommend that you never use the little screws that come with tuners. Throw them away and use the same size Sheet Metal Screws. You will save yourself much grief.
By the way, stainless steel Wood Screws aren't really any stronger or harder than normal plated steel Wood Screws. They are normally made from 18-8 stainless, which is just a soft mild stainless, about like 1018 mild steel. I don't know why people think that stainless steel is so much stronger than mild steel. Some of the special alloys are, but not the alloys used normal parts.