Replacement of the thermostat in my 2003 Mercury Grand Marquis

There is no correlation between the thermostat and burning oil. This explains what causes oil to burn more than normal.

https://www.700r4transmissionhq.com/mercury-grand-marquis-burning-oil/

https://www.google.com/search?q=2001+Mercury+Grand+Marquis+is+burning+oil

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Could very well be the sensor (thermostat). No idea where yours is located, but parts store location chart may help, or you could buy a shop manual.

Well code 128 is for a thermostat problem in the fact the engine is not reaching normal temp. If you have heat the engine is warming up but still may not be getting to normal range. Normal is 190 - 210 degrees.
I would replace the thermostat first and see what happens.

ALL YOU HAVE TO DO IS BUY YOU AN NEW THERMOSTAT GASKET SET AND SOME HIGH TEMP GASKET GLUE REMOVE THE TWO LITTLE BOLTS PUT THE NEW THERMOSTATS AND GASKET AND GLUE GOOD AND MAKE SURE NOT TO OVER THIGHTEN IT BECAUSE IT WILL STRIP EAZY THAT ALL TO IT I HAVE THE SAME YEAR MAKE AND MODEL

did everything said to do put new heater core in and a new thermostat instill no heat

It sounds like thermostat (in coolant system) needs to be replaced. They only control you have for temps are the dials on dash, but if coolant thermostat is sticking you're limited to what it outputs. Replace coolant thermostat and you should be "good to go"

check coolant level, you may have air trapped in the system.
Also, is car engine warming up to normal range on gauge? IF not, change thermostat.
Loosen the radiator cap when cold. start car, let engine warm up and see if you get a burp of air out. Fill radiator as it cools.

Locate the water hose at the top of the radiator and follow it to the top of the engine block. There you will find the thermostat housing. Remove the two bolts and remove the housing. Pay attention as to how the thermostat is sitting in the housing. Lift it out and put in the new thermostat the same way. Make sure you replace the thermostat housing gasket. Reassemble the housing and add anti-freeze until full.

check the thermostat values if correct to manufacturer,

Start by checking your coolant (antifreeze) level. This will cause this problem.