5 1/2 month Shepard Pup NOT housebroken!!?

having trouble housebreaking my little guy and I need HELP! He is a rescue he was on the way to the shelter at 5 weeks & a big part of the problem is that i was not prepared for a pup when I took him home. Very irresponsibly I had no crate and no means by which to isolate him excpet in my bedroom which also has wall to wall carpeting. We used puppy pads in the begining which i know is also created confusion about the appropriate potty areas. Needless to say we're five months down the road and now my carpets are destroyed & I can not get the smell of urine out of my apartment. I'm certain that because I can't get the smell fully out of the carpet it is confusing for him but the worst part is he gives almost no indication that he needs to go out. Overall he is a wonderful dog very mellow temperment not rambunctious follows commands walks well on a lease doesn't run off if he's off the lease he comes when he's called almost never barks etc. I don't think that there is a problem with leadership but rather that he just doesn't have a clear cut understanding of where the appropriate place to go is. Occasionally he will whine very subtly to go out but much of the time there is no indication. I will just hear him urinating in the next room or will walk into my bedroom for a second and come out to find he has eliminated on the hallway floor. We might have just gone out! I am trying to be as consistent as possible and offer lots of praise when he does go outside but honestly it seems to have gotten worse. There seems to be no schedule to when he has to go and I've tried limiting water and food to specific times makes no difference. I am ready to replace my carpets just to eliminate the scent problem. I am at such a loss! The strangest thing is that to the best of my understanding he has not had an accidents while I am at work. We go out as soon as I come home and it never seems like he's had an accident in the house while I was away but when I am home he often goes inside. I just don't understand. Anyone who has any info please I could so so so use your help