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Just google "2006 Tahoe towing capacity"
http://www.edmunds.com/chevrolet/tahoe/2006/features-specs.html
7,800-pound towing capacity, when properly equipped
You should have oil cooler and transmission cooler, if you have factory tow package. If you are not the original owner or you just aren't sure, check the RPO codes on Internet (they are on sticker in glove box). Also, check for the coolers. The trans cooler looks like a mini radiator in front of your actual radiator. The oil cooler is a vertical, finned unit, also in front of radiator.
They certainly don't make it easy to get a peek at the radiator, but yes, it appears that there is a transmission cooler in the front.
I got a pic of the RPO codes, how do you tell what each one is for?
How heavy is the camper you're looking at? I pull a 7000# trailer with my 01 with no issues. I have the tow package also, but opted for a bigger trans cooler that I run as a secondary cooler plumbed in AFTER the normal radiator cooler. I also had a corvette servo installed and the line pressure bumped up a little to firm up the shifts. The combo of those things work great, though the 5.3 is a bit undersized for some of the hills around here. The hoe does it, but you can tell it's not happy about the hills. Good luck!