Right turn signal not working properly?

Okay, so, I found the flasher relay and it seemed to be not pushed in all the way so I pushed it in and nothing changed its the passenger side turn signal only that's not working even after bulbs are new and fuses are good also the emergency flashers work on both sides all lights work all around accept the right (passenger)signal and its dash light comes on and goes off on its own but never blinks and when the switch is put to turn it off it won't go off until it wants to the pics below show you that all lights work other than that 1 light. So, now can you give me your thoughts on where to look next or do you think it's the actual switch on the steering wheel.

Okay, so, I see where the flasher plugs in at and the wires coming from it. So, is it a part I can buy and if so, how hard is it to replace? Also, would the hazard lights still work if the flashers bad cause both sides blink when the hazards are on.

I thought I should mention that regardless of if I hold the signal switch up or I f I click it to get the light to blink it doesn't, it gives no light or any blinking on that side other than when the hazards are on and this is both in the front and the back of the car however the signal arrow on the dash will come on and stay on as well and will go off on its own as well.