Be Careful! Even though the "10xxT" series is listed, you have to pay attention to the Wattage! According to HP, the Pavilion p6803w is only designed to handle up to a maximum 95W Processor. Only about half of the 1000-series is 95W, the others are 125, and even 145W and may quickly blow the VRMs (voltage regulators), if they even let it run at all. I would not take that chance on a 5-7 year old motherboard unless you want an excuse to buy new.
Wikipedia has an easy chart of all the processors and their wattage.
https://en.wikipedia.org/wiki/List_of_AMD_Phenom_microprocessors
That 1100T you mentioned needs 125 Watts; the 965 Quad Core that @kanewolf mentioned is either 125 or 140 Watts; both Waaay to power hungry for that PC.
At a cursory glance, it looks like the best processor you can do is X6 1065T.
For a 95W processor, you'll be o.k. with any aftermarket cooler made for the AM3 socket, as the socket on other motherboards can handle up to 140W processors. Something like the Wraith Silent or Noctua L12 that has a horizontal fan will get you around the case-height clearance issues.