Once again, not to sound like a know it all - but I was the person that wrote the technical service bulletin for that particular engine with that particular problem.
Here is the story:
In 2000 - my brother bought a new Z/71 Silverado ex cab pick up truck with the 5.3
It had so many problems that the GM rep came to the dealership on several occasions to try to rectify the situation.
The one problem was oil consumption.
The engineer at GM tried to say that because the engine sits in the engine bay at an angle and because the PCV valve was located in the wrong place, the PCV valve was sucking up oil when the engine was off - because the oil puddled in that area of the head and when you started up the engine - it would suck the oil into the line and it would suck a little oil.
Well there was one problem with that theory - the truck was driven 81 miles each way to work every day and it got to the point of where it used a quart of oil every 3 days.
They even Blue Goo'ed - Permetex'd it shut and it still sucked oil.
In the end it was determined that the valve seals were at fault and because of their design and because of the materials used in the valve guides that the valves wore the guides and the valve seals did not seal and when the intake valve opened - a rush of air out of the crankcase entered into the fuel stream with the air from the intake manifold and the end result was that it would funnel a quart of oil every 500 miles - or one quart of oil every 3 days.
So I played their game and I payed for an oil change and they did a oil consumption test. After the 5th quart of oil, they tore down the engine and rebuilt the heads. Technically - they should have replaced the heads - but I know that they wanted to find out what the real problem was - and not have to tear down the motor again in 30,000 miles when it started to funnel oil again.
The heads in Question were MADE IN MEXICO. They usually have a Tri Angle inside of the head and they also will sometimes say Made In Mexico - right inside of the head. I'm not sure - but I believe that they used a different - lower compression style head on the 6.0 engine and that they are not the same casting.
So the solution is to find a good engine shop. Pay about $1000 - to $1500 to have the heads removed, resurfaced, hot tanked, all the valves and guides removed and replaced and new valves and have them reseated and use a newer style valve seal that will not leak.
Now not to hi jack this post - but this is not the end of your problems.
The next problem that you are going to have is that because the engine consumed so much oil, the catalytic converter is soon going to fail.
It's not going to be today, it's not going to be tomorrow. But I would guess that somewhere along the lines of about 125,000 to 140,000 - the engine is going to loose power and fuel consumption is going to go up and you ar going to scratch your head and wonder why it won't run right and why it won't go up a hill no more.
Like a dummy, I removed the guts from my converters and I had to buy a scan tool - because once a day, the check engine light would come on and say Catalyst efficiency below normal threshold. When you remove the cats - the computer goes into the limp mode and it still won't run right and will suck fuel and the torque won't lock up right all the time.
So if all this scares you - your best bet is to get rid of it now because once the gremlins comes out - you are going to throw good money after bad at it. It's sometimes cheaper to just trade up and make sure that you don't get another one with the same problem as you have now.
The 2002 Avalanche Z/71 we had, did the same thing at the same time with the same oil ( Pennzoil) and with a 3,000 mile oil change interval and always ran good filters and never beat on it.