'99-'00 chevy blazer

I would think it would be better after you change the head gasket and drain the old oil from the system. Then check the new oil and depending of how much water you are seeing in the new oil you may want to flush the engine. I suspect it will not be necessary because the time it takes to change the head gasket will also allow for the excess water to drain to the oil pan.

Hi Wendy, hope I am wrong but it sounds like a cracked head or blown head gasket.
If the engine hasn't been started in a couple hours pull the engine oil dipstick and look at the oil level. If the oil level is over the full mark you likely have water in the bottom of the oil pan. Milky/semi-white oil on the dipstick would indicate the same. If the oil looks good and at or below the full level look in the water overflow for the radiator. Any sign of oil or milky looking foam is an indication of oil in the water from cracked head/block or blown head gasket. If both the water and oil look good start the engine, let it warm up and look to see if you have water coming out of the exhaust pipe. It is normal to see some water come out of the exhaust after starting a cold engine due to condensation in the muffler and catalytic converter but a healthy engine should have only a slight amount of water vapor coming out of the exhaust. Anything more would indicate a cracked head/block or blown head gasket.

Yes, muddy coolant means there is oil in it. Check it by dipping your finger into the coolant while it is cold and rub your fingers together. If it is oil, you will feel it right away, while rubbing your fingers.

If you have the 2.7L engine, you may have a bad water pump. The water pump is turned by the timing chain. When the water pump on those particular engines go out, tension on the chain goes away and retards the camshafts bad enough for it not wanting to start. Pull your oil dipstick out and see if you see signs of water in the oil. If there is, this is your problem. When the water pump goes out on those engines, it'll drain water straight into the front of the oil pan.

you my have thick smoke coming out of your tail pipe or oil leaking out on to your engine

a leaking intake gasket or hose is usually the problem but also could be the water pump or head gasket about to go. keep an eye on the oil if it starts to look like chocolate milk you have a head gasket that has gone bad

Your engine is overheating, which isn't directly related to oil pressure. I would take the S10 to a reputable/trusted mechanic and have him perform the following tests.
Compression Test Leak Down Test Coolant System Pressure Test Inspect Oil for Moisture (you can do this one, remove the dipstick, does the oil on it look like milk? If it looks like milk it means there is water/coolant in the oil, a sign of a blown head gasket. However not all head gaskets dump coolant into the oil when they blow) Inspect Clutch Fan for proper operation
The mechanic will know what those mean, It sounds like your head gaskets may be blown, which is why the engine would be overheating. Also if the Clutch Fan isn't working properly, it won't pull enough air through the radiator to keep the engine cool at an idle or while sitting still.
If the Head Gaskets are blown, it doesn't necessarily mean you need a new engine.
One other thing to note, when an engine overheats, the oil breaks down and no longer works as good as it used to.

white smoke means your getting water or antifreeze in exhaust, maybe a blown head gasket, best way to check is give it a compression test

there are a few reasons for the engine to smoke. first determine the color of the smoke, white smoke means its burning water...possible bad head gasket or cracked block or cylinder heads. If the smoke is black...means the fuel mixture is to rich, getting to much fuel. If the color is blueish...it is burning oil, meaning the oil rings are worn, and or the valve seals are worn as well. if it smokes when you first start it, and then stops after a few minutes, once the engine is warmed up...probably bad valve seals. if it continues to smoke it is more likely the oil rings. Valve seals being cheaper and easier to repair, where as the oil rings means the motor needs to be rebuilt or replaced.

It may be your heater core. the connections are plastic and do brake. Do you get hot air from your heater or defrost? Can you see the leak? Is it for a head-gasket? How's it running? Is it rough?