I have a setup like that 105 levers and 10 speed FD setup.
NOTE: 105 has a trim click that produces a short movement on first short movement of lever the second will move the FD again and move chain.
I removed cable make sure lever is all the way down to small ring push small lever right till it stops clicking with chain on small ring set limit screw just till the left inner side of FD clears chain about 1 mm clearance then with your fingers get chain on large ring push FD with your finger and set stop screw same but with clearance on right side this time.
Now put chain back on the small ring make sure lever is clicked all the way down again screw in (down) barrel adjusters and back out 1 turn so you have something to play with if needed, now pull cable tight and lock it down.
Take up slack with barrel adjuster try shifting lever should click and shift if no click the stop screw is not allowing the FD to move up enough turn stop screw 1/2 turn and try keep this up till it shifts and has a solid click.
Now shift down it should drop if you find it later dropping to far down and off small ring turn in stop screw 1/4 turn till it stops dropping, if you do this check cable slack 1/4 may not make a difference but more may so check it.
When done correctly you will also have a trim click before it moves chain to next ring with 105 levers.