a six foot fence with posts should be the minimum of 2 1/2 feet in the ground. One foot in any soil will not work.
I would do this, since the ground as you say is very wet. dig your hole about 2 1/2 feet deep, set a brick or something like that on the bottom. put the post in the ground and pour a bag of Fastcrete quick set into the hole without any water. Support the post and let the surounding ground water soak in and set the post. This should be set fine by the morning. In normal conditions I put in the fastset dry and then add water to the top and it will set in fine.