Three possibilities come to mind.
The first possibility is a problem in the governor circuit. Luckily, this problem can be easily checked with a pressure gauge. There are four pressure ports on the passenger side of the transmission. The port that is about 3 inches higher than the other three is the governor pressure port. The pressure in this port should be approximately 1 psi per MPH until the converter locks up or the transmission shifts into overdrive. If the governor pressure does not increase with vehicle speed, then the transmission will not shift correctly.
Another possibility is a faulty transmission temp sensor. The PCM controls the shift points in your transmission. When the transmission fluid is cold, the PCM makes the shift points approximately 3-400 rpm higher than normal to help heat the fluid. A faulty temp sensor could be stuck in the 'cold' shift pattern causing your delayed shifting.
The last possibility is what I consider the most likely case. The lip seals on the front clutch are notoriously bad in the 518/618 trannies. The drums were updated in 1999 to take care of the problem but some of the older trannies ('97 & '98) still made their way into new trucks even until the 2001 model year. You can check for yourself what year your transmission is if you drop the pan. The valvebody is held on by 10-7/16" bolts. Three in the front, four on the side and three in the rear. Near the last rear valvebody bolt is a casting emblem with the model year in the center.
Since the front clutch is also applied in reverse, you should notice a delayed engagement into reverse when the transmission is cold as well as the delayed shift into third gear you just described.
The problem can be fixed with about $20 worth of parts. A valvebody and converter package will NOT fix this problem although several vendors would be happy to make a sale. You might get told that you need a new transmission because this too would solve your problem, but that is surely an expensive way of solving a $20 problem.
The problem is that there aren't enough people in the transmission industry capable of diagnosing a problem correctly. Most people simply throw parts at a problem or rebuild the transmission hoping the problem will go away instead of finding the root of the problem and solving it.
A DTT update kit was specifically designed to solve this problem. And the problem is so common that DTT no longer sells a Torque converter or valvebody without first addressing this problem.
Best of luck to you,
Chris