95 buick lesabre won't go unless running for awhile

It's possible in this case that the flushing of fluid may have caused more harm than good. If a transmission has never been flushed but only drained and had it's fluid and filter replaced (which only changes a small part of the fluid) there sometimes are tiny bits of metal that work their way into intricate places. Such as internal seals and gears. When a flush is done on a transmission which has never had it priorly done sometimes it can cause an internal leak or gear slippage due to these little bits of metal being removed. The manufacturer of a transmission flushing machine states not to use it on transmissions that have not had it used on them for a regular maintenance service. By regular they mean (for most vehicles) every 50 to 70 thousand km's. That being said if your new but used transmission was not regularly maintained with flushes the flush done on it may be the catalyst to the problem. I suggest since it was a used transmission to return it whomever it was purchased and procure a new (or new used) one. If this is a problem due to place of purchase and the seller refusing to take back (hard to do sometimes with private sellers) a now faulty transmission, you could check your local laws about purchase and return. Possibly you could legally force a return. For instance any used car can be returned by law in Canada within 90days of purchase if it breaks down for some reason, it is law that they take it back and return to the purchaser the funds of purchase. This law does not pertain to parts sold however, but a good lawyer should be able to make the case.

That's why I never changed my after 130,000 . Had to let it warm up 15 or 20 minutes when it was cold. Made to 197,000 miles by letting it warm up a couple of minutes then slowly shiftin down to first gear back to drive and away I go.

Waiting for engagement can be caused by internal seal leaks, however, there is a check valve in the transmission cooling line that prevents the fluid from the converter from drain back into the transmission. If the check valve is not performing or missing, this will cause a hesitation in engagement.

If you are not well versed in mechanics, the only way for you to check this would be when parking for the night, shut OFF the vehicle. Remove the transmission dipstick and mark where the fluid level is. The next time you use the vehicle insert the stick before starting the vehicle and see if the fluid level is MUCH higher. If it is this could be an indication that the check valve is not present or not working.

May have used the wrong fluid or may have low pressure due to bad or failing seals or contaminated fluid ,what happened to the previous transmission how did it fail? did you replace the convertor?