Are both of my SSRs broken or am I doing something wrong?
You don't have enough of a load on the output of the SSR.  

Most SSRs contain a snubber network that consists of a resistor and capacitor in series and then connected in parallel with the output terminals of the SSR.  The capacitor allows some leakage current to flow.

This leakage current is enough to illuminate neon lamps and will even cause some 120 Vac strobe lights to flash periodically.  Connecting a small incandescent lamp (5 Watt Christmas bulb) across your load terminals is usually enough to swamp out that leakage current.

Note: the snubber network is required for the triac inside the SSR to not false-trigger when certain conditions are just right.  But the snubber does introduce its own set of problems.