Transmitting 12v DC over a long wire
The wire is probably 24AWG (0.51mm diameter).  At a length of 10m you actually have 20m of wire (there and back again) (20m is 65.6 feet).

24AWG wire has a resistance of 84.1976Ω per 1000m, so 20m should be a resistance of around 1.684Ω.

As you can't (or don't know how to) give a current value I can only give some examples.

At 1A the voltage drop across the wire will be (V=R×I) 1.68V, so the voltage at the far end would be (12-1.68=) 10.32.

At 2A the voltage drop would be double that, so the resultant voltage would be 8.64V.

At your guessed 3.5A the voltage drop would be 5.894V, leaving the 12V at a paltry 6.106V.

And so on.

The maximum recommended current for power transmission on that gauge wire is 0.577A.  Any more than that and the self-heating of the wire could overwhelm the integrity of the insulation, and even cause the wire itself to melt.

At 0.577A the voltage drop would be (0.577 / 1.684 =) 0.343V, so the voltage at the remote end would be 11.657V.

The answer?  Use thicker wire.  The thicker the wire the lower the resistance, and the lower the losses it causes.

For 3.5A the recommended minimum gauge wire is 16AWG (~1.3mm copper diameter).