I had a 2003 TB EXT with I6, 3:42 gears. We need to know a bit more about the truck. EXT or not? Gear ratio, look in the glove box for the RPO sticker. Three axle ratios were offered for the GMT-360/370 platform. The 4:10s in a TB with an I6 is pretty rare.
GU6 = 3:42
GT4 = 3:73
GT5 = 4:10
Once you know this information you can cross reference it in your owners manual in the towing section to find out your max towing capacity.
The I6 is an excellent engine. My max towing capacity was 4099#. I towed a 3500# hybrid trailer with it, once out west in the Rocky mountains. Wonderful towing experience. Later for one season we towed our current Jayco X20E with it. 4750# GVWR. Adequate but not stellar. I would not have towed it in mountains. The TB had 160k on it so it was an easy decision the next season to upgrade to the Sierra in my sig.
That said all GMT-360/370s have the towing package standard. There is a built in transmission cooler that is more than adequate for most towing. I didn't add one until I got the bigger trailer. I would change your transmission fluid and make sure to use DEX VI instead of the factory installed DEX III. Dex VI is a synthetic blend that came out in 2006 and is recommended for older transmissions. It runs cooler and is a better fluid.
Another thing to consider is payload (found on the yellow sticker in the door) and tongue weight. With a WDH the max tongue weight is 920# as I recall.
If you keep the GVWR of the trailer 1000# under the max towing capacity of the TB you'll be fine in all respects. More than that and you'll be disappointed with the towing experience.