I'm no expert on 105, but here goes.
H/L adjusters - often, these are marked H and L of course. Presumably not in your case. I usually just peer inside the derailleur and hope to see which is against the stop. Or, put it in top or bottom. One screw will be hard to move - that's the one that does the end where the gear currently is, because it's against the stop, whereas the other is currently floating free. Or, in that position, try moving both screws and see which one causes the derailleur to move. Or if desperate Google the manual.
Yes, if I remember rightly, my Shimano shifters have more than two positions. Some ability to trim is needed, as the chain is in a different position when it runs to the inner versus the outer sprocket. Campag always had far more trim, which made tourists (with their wider gear ranges) often prefer it.
I've got adjusters where the cables leave the shifters. There's usually at least one in the system somewhere.
It may help to know which 105 you have - at least how many sprockets on the cassette.