Dual power supply with two different amperages
If you want 3amps from the +5V regulator, the input voltage to that regulator should at every moment remain above +5v, plus some more because the regulator needs some overhead.
The data sheet for LM1024 says that 1.5V of overhead is required when 5A flows - let's use that figure. So input voltage to the regulator must always be above 5+1.5=6.5V
When 3A flows, the diodes charge up the big capacitor in a rather short time period near the peak of the 60 Hz wave...actually it is a 120Hz waveform because you're using a 4-diode full-wave rectifier. So assume the capacitor charges all-at-once, then discharges @ 3A for a period of 1/120 seconds....0.0083s
You've measured the unloaded DC output voltage near 14V. This seems like plenty for a regulator chip that only requires 6.5V at its input. But 3A pulled out of the capacitor depletes its voltage rather quickly...too quickly. You need a big reservoir of charge, and that means a big capacitor.
You can work out minimum capacitance required from \$I=C \frac{dV}{dt}\$ where I=3A, dV=(14-6.5) and dt=0.0083 seconds. Find C in units of farads. Your C of 1000uF is too small.