Will a step-down module increase amps and keep power supply steady on other 'branch'?
Power = voltage x current
Calculate the power available:-
12 V x 1 A = 12 W
Calculate the power required:-
12 V x 1 A = 12 W
5 V x 3 A / 0.8 (estimated converter efficiency) = 18.75 W
Total power required:-
12 W + 18.75 W = 30.75 W
Clearly you not have enough power to run both devices, nor even enough to run the 5 V device at its maximum rated current draw.
So what do you need?
30.75 W / 12 V = ~2.6 A
You need an adapter which can deliver at least 2.6 A at 12 V, preferably more.