Air carrier First Officer reported confusion of ATC clearance at DCA and the taxi direction sign position resulted in wrong turns and taxiway incursion.

Narrative

After excessive delays throughout the day (which caused us to nearly time out on our next leg by 4 minutes after our duty extension); the Captain made a normal landing from the River Visual 19 approach into DCA around XA:00am. While decelerating on Runway 19; tower instructed advised that our gate was occupied and to turn onto Runway 22 then A and hold on B. The Captain began left turn onto Runway 22 (he later said was because Runway 22 begins to the left and ATC never said which direction the turn onto the runway was to be made). I immediately told the Captain we needed to turn right onto the runway (toward taxiways A and B) and he began to turn right after a 45 degree turn to the left had been made. As we were coming back to the right Tower asked us where we were going and I told him we were correcting and taxiing A to hold on B. Then as we were taxiing on Runway 22 toward A; the Captain began to turn right toward C. Then taxi direction signs at the angle they are positioned did make C look like it was A at night. Before fully entering C; I noticed small cones blocking taxiway C and advised the Captain to be careful with the cones. We realized this was Taxiway C and began a left turn back onto the not in use Runway 22. Then the Captain turned right on A and we held on B until cleared to our gate.It had been pointed out to me on a previous trip how the taxi signs for crossing Runway 22 are easily mistakable based on the roughly 45 degree angle with the runway. I believe this; combined with possible fatigue from our long day were contributing factors (even in ZZZ we were roughly 3 hours behind schedule with weather delays and after preflighting for this flight were told we needed to change planes and quickly started over at a new gate).