Check engine light on, auto transmission shifts gears abruptly

The Check Engine Light is on? Did you read the DTCs.

The second link has the answer, the first link says it is a known issue. It is times like this you need the Dealership or a good auto electrician.

Transmission won't shift to third until temp rise

ClubLexus https://www.clublexus.com > forums > 713188-transm...
Apr 1, 2017 - Transmission won't shift to third until temp rise ... The solution is to drive slow and keep rpms at 2.5k and 3k at max till the engine and transmission warm up.
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Owner of Lexus RX300 with 182000 miles. Recently the ...

JustAnswer https://www.justanswer.com > lexus > 3wsxt-lexus-rx3...
Aug 22, 2010 - Recently the transmission won't go into overdrive. Stays in 3rd gear like when you press the o/d button to do so. When car is cold (live in ...
1 answer · Top answer: Hello and thank you for using just answer! The check engine light is on correct?

https://www.google.com/search?q=Lexus+RX300+will+shift+into+overdrive+once%2C+and+after+slowing+down+it+won%27t+go+past+3rd+gear

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Jorge, as richie46 recommended check trans, fluid level. At higher mileages,150,000 the automatic transmission may not shift correctly. This can be caused by the throttle position sensor out of adjustment or a shift solenoid needing to be replaced. Typically the transmission does not need to be completely overhauled.
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I don't think overheating of the engine has much to do with the transmission, either automatic or manual. If you have an automatic transmission, did you check the fluid? Sometimes low fluid stops certain gears from changing, as the torque converter doesn't have enough fluid to pressurize the servos to engage the clutches.

I would go ahead and get the engine computer checked for trouble codes just to be sure. Most auto parts stores will check for free.

Check your input or output sensors on top of tranny. Check engine light?

run the fault codes to find the reported fault
engine lights report on engines and transmissions
could be actuator or vacuum to actuator, dirty transmission oil or pump pressure problems

the check light has a sensor fitted in the engine, this inhibits the gear-box for a reason, there must be a problem somewhere, thats why the light is on, it trips the sensor to stop over use of the gear-box and engine,until the problem is rectified, so as to reduce the chance of damage...

your pump inside the transmission might be bad or governor isnt wronging right..but its all about the tranny here...

The computer does not know what gear it is in. You need to replace the transmission position switch 8-96017-511-0

I had the same problem in my 96 f-350. The transmission has slipped recording a slip code in the transmission. You can pull the codes and get it. I shifts into a safe mode that increases the pressure inside your transmission as part od a fail safe. Plain and simple, you have to have the tranny rebuilt.