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The first thing you'll need to do is have the powertrain control module (PCM) scanned for fault codes. You my have a deeper problem, but it sounds like you may have a bad output speed sensor on the trans. This is the sensor on the left side of the overdrive unit at the back of the trans. It tells the PCM the output shaft speed of the transmission which is then used for governor pressure calculations for the electronic governor. When the output speed sensor signal is missing there isn't any way to control the governor and it will normally stay in first gear without attempting to upshift.
If you have a voltmeter you could do a quick test of the output speed sensor before having the codes read. It's a two wire sensor, check the continuity between the pins of the sensor. If it's an open circuit the sensor is bad for sure. It runs about $20 and will be in stock at any parts store or dealer.