There are other simple ways to confirm a screw is bottoming out. I can usually tell just by the way it torques up, but wood density and compressibility can vary. So, tighten your screw to your favorite setting and make a witness mark with a Sharpie on the screw head and on the escutcheon at say 12 o'clock. Remove the screw, grind or file off about a 1/4 thread and reinsert and tighten. If the witness mark on the screw head rotates past the mark on the escutcheon with the same torque value, probably by about a 1/4 turn, you know you are/were bottoming out. Another simple check is dropping a small lead ball, like bird shot, or the very fine shot used in .22 shot shells, into the hole. Tighten and look for the smashed lead ball. This will also tell you how much to remove from the screw. Thin lead wire will also work well, if you have some. The lead wire is the gunsmith standard used for determining the precise amount of headspace in a chamber when a go-gauge "goes" Close the bolt on the wire behind the gauge and measure the resulting thickness.
Since both screws exhibit the same behavior, I personally feel it's much more likely that either the hole is shallow, or the wood and/or escutcheon is compressed or too short. If shortening both screws fails, then thread locker is warranted. Unless you have removed and replaced that action screw 100s of times, I doubt the hole is oversized. The screw will also have a distinct loosey-goosey feel when screwing it in. IF it is oversized, though, any machine shop can make a slightly oversized screw on a lathe, but don't even think about going there yet.
My two cents...probably worth less than that.