Like you I've only been over during the TT. We have always camped at Peel. There is water and electric hook up and the showers have been extended (in the gents anyway) with the old 4 showers now being supplemented by about 6-8 more out the back. They also have a TV room. There are 4 indoor sinks for washing up and there is an electricity socket in the wash room where people charge their phones. There is WIFI on site but you do sometimes have to be quite close to the building. The field is pretty big so I think you could let the dogs loose depending on the number of other campers.

The small car park at Fenella Beach next to the castle I think says no overnight parking, but I'm pretty sure vans have been there overnight. You could take the dogs up Peel Hill and along the coastal path from there. There is parking at the far end of the prom too but I don't know what they might think about camping there.