This problem comes up constantly. The 1A rating means almost nothing. Don't count
on loading the v.reg to 1A without it getting rather hot.
The smallish SOT-223 v.regs are only good for about 0.3-0.5W dissipation. This means
for Vin=12V, Pd = Vdrop*Iload gives a measly Iload = 0.5W/(12V - 5V) = 71 mA or so.
Slightly better if you factor in the series-diode drop. You can figure Iload out for other
values of Vin.
Larger v.regs, like D-PAK, are somewhat better.