There are a couple possibilities:
Check for "blowby", that's when exhaust gas leaks past the pistons, into the crankcase. If the engine has blowby it will be much worse when the engine is under a load.
Check the vacuum lines, especially the lines from the transmission. You're looking for oil in the lines. Sometimes a broken vacuum diaphragm will allow oil to be sucked into the engine. In this case it could be that the diaphragm leaks oil only when the engine is pushing the transmission hard.
check the piston rings but that a typical thing any 350 dose check the piston rings but that a typical thing any 350 dose
replace the cylinder gasket and check for cracks and replace the head gaskets
see if your spark plug are wet, change oil, with new filter.
Have the head gaskets checked and intake manifold gaskets checked for leaks. The 3.4's are known for head and intake leaks! Or a bad thermostat.
Head gaskets are changed on a 97 Chevy Blazer by removing all engine accessories, taking off the intake manifold, removing the valve covers, and taking out the retaining bolts. The heads can then be lifted, the gasket removed, and a new gasket put in place.
Running the engine hot repeatedly.
Probably a gasket leak. Could be the intake gaskets or the thermostat gasket.
have your intake gaskets checked common with rodeo
They have problems with yhe intake manifold gaskets. Felpro has a improved set that do not leak.
have the heads pulled off and checked for cracks have them resurfaced and sand the block until it looks new. then replaced. gaskets with several layers of sealant on them.
"Engineering improvements were made to the Chevrolet 2011. Horsepower, torques, and head gaskets were made more powerful. You should test drive both of them."
The heads don't necessarily need to be resurfaced when you change the head gaskets unless you think they may be cracked. They should be resurfaced every time the gaskets are changed, but you can get by without resurfacing them.