ULN2003A output not going to logic low
You have the diodes backwards, which will likely damage the ULN2003A and the diodes. They are supposed to be reverse-biased when the relay is energized.
You also don't need them, there are diodes in the ULN2003A, which you have correctly connected to +5V via the COM pin (well, it will reduce the loop area a bit if the diodes are very close to the relay, so they're not entirely useless).
It would be a bit more clear if you followed convention in drawing the schematic and had the supply rail at the top.