Can i drive my mdx with the check emission light on

Question edited for clarity.
Question moved to model category.

You need it scanned with a pro grade scan tool.

https://www.google.com/search?q=2007+Acura+MDX.+SH-AWD+light+comes+on+solid

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sounds to me like a BAD CO2/Lambda Sensor Good Luck Mike L.

has anyone checked for fault codes

Start with the first mechanic who replaced the battery and have them check the charging system. (Alternator) They should have done this when changing a battery, to rule that out as a problem in the first place. Explain the problems you have been having, as a bad charging system can cause all sorts of electronics problems and have them check the engine codes and have them explain what they are.
You could also go back to the muffler shop and tell them their sensor replacement did not fix the problem and have them correct their work.
Diagnosing Alternator Problems EricTheCarGuy

Exhaust and Emissions Auto Parts for your ACURA MDX

this may or may NOT help please let me know.

2001 Acura MDX Repair Problems Cost and Maintenance

then means its busted. and not a random glitch.
why not scan it and get the DTC errors... ?????
always scan it first. write down the DTC number Pxxxx
then reset
drive and see if it fails again , if it does, it's REAL.

P0400 Acura - Exhaust Gas Recirculation System Leak Detected

Hose, gasket, or egr has gone bad

SOUNDS TO ME LIKE A BAD CO2/LAMBDA SENSOR

Depends - it could be a reminder to have the oil changed, or a problem with the emissions and engine which would need to be diagnosed.
Your owners manual should tell you what the message is for.

SCAN VECHICLE CHECK FOR TORQUE CONVERTER FAULT OR TRANSMISSION FLUID PRESSURE SOIENOID. CODE.AND CHECK TRANSMISSION FLUID LEVEL