The card is not 22 amps, 22 amps for a full system.
As long as the system fits into that power envelope, you are good to go.
and Watts = Amps x Volts
so 19 x 12 = 228 watts. That is not all that bad for a 300 watt power supply.
For the real kicker..... I have a 300 watt power supply with 22 amps and am running a GTX 650 ti
Had a 5770 on it for at least a year too. Its all about how much power the rest of the system takes.