Dear brother Sprite I am not an expert but have traveled on thatta - sajawal route and stopped over the bridge to have a look at the Indus mostly it is sand but these days I am quite certain that it must be full from bank to bank with all that flood water draining down. I suppose to witness the actual mouth of Indus draining in Arabian sea you will have to travel towards Keti Bandar (to be precise Goth Chelaram) search for it on google maps and you will get as close to the mouth as possible, it is all mostly katcha area, you can also take a boat ride from Keti Bandar towards the actual mouth.
One a side note, flamingos do reside in these areas all year long if you wait few days more (maybe 15 to 25 days) waterfowls, sand pipers and other migratory birds could also be seen in these areas I guess but if you are following the flood it is an altogether different story.
Security is not a very major concern, however, reports of street crimes after dark are there.