Easiest way to check if uln2003a is working
You are misinterpreting the datasheet. It is guaranteed to be able to sink 300mA when 3.0V is applied. That's a functionality specification. Below 3.0V (while still above ground) it is not guaranteed to sink 300mA. It might do something with a couple volts in, but not likely with 1.5V and no guarantees below 2.7V for performance with input voltage- and that's with 2V drop at the output.  

The maximum input voltage is specified as 13V. 

Try this out with a visible LED. Use a 1K series resistor and a 5V supply. Apply 5V to the input (or 3V). You can also damage the ULN2003A by leaving out the resistor. 

It is fairly unlikely you have damaged the chip by soldering to it, however you have a good chance of damaging it by leaving out the series resistor.