Hello Dhruv
If you get seat in any of the IIT than you have to report to any of the IIT ( same or other than you allot ).
And if you got nit or IIIT than you can go in any of the nit or IIIT for verification ( it is not josaa choice ; it will be your choice where you want to go for verification ).
You have to verify documents only once after that you don't need to go anywhere ; you have to just go for admission in alloted college after all rounds and in the first time you choose one of the options from float ; freeze and slide.
So good luck