Is the transmission the same type as the OEM?

Yes. Only difference is that it came from a 1994 Lexus SC400

My concern is there is a difference in sensors or shift solenoids and the computer can not communicate properly with it.

Have you checked for trouble codes?

Good question. My impression is that if the computer was the problem, the issue would be more consistent; meaning, why would the difficulty with the transmission shifting automatically only occur after the car hasn't been driven for a couple of days and resolve itself shortly after being engaged. Also, why did the shifting issue only occur after the flushing process. Lastly, what changed between the 1994 and 1995 models? Have a good day.

Our database indicates that there are no changes between 94 and 95 models.

From the symptoms described, the most common cause for such faults lies with the valve body, usually related to the solenoids.

There are manufacturers that strongly advises against flushing transmission because it can create problems when debris are stirred from somewhere to be lodged into orifices causing shifting problems.

If there is a problem with the solenoids etc, the trouble codes would tell you which is not working correctly. Some solenoids might stick and work itself free after some running.

Such causes would not be consistent.

If the debris is chemical in nature, are you aware of any solvent or other chemical solution which can resolve such a blockage? And, why does the blockage disappear once the unit is warm (or after the transmission is engaged for a few minutes)?

Have a good day.

Usually these are bits and pieces of metal filings or clutches and there are no chemical that would clear them.

It might not be blocked orifices that is causing the problem. It could be stuck valves or solenoids that worked themselves free after warming up.

Under such circumstances, your best bet would to have the valve body removed and serviced if there are no trouble codes.