Seasonal bored well

We have an 80ft. deep, 2ft. diameter bored well. It is all clay except a 1ft. area of sand where the water comes in at 42ft. From December to June/July, the well is so full that you can touch the water. Then, it suddenly starts draining out and the well will go dry in about 2 weeks. In trying to calculate the rate of the water coming into the well, we have pumped the well dry. Then, three days later, there was only 105 gallons of water in the well. We don't know whether we have an extremely low yeild well, or whether it is just draining out that fast. We considered that it was back-draining through the 1ft. area where it comes in at, but we would at least have water from there down to the 80 ft. mark, but we don't. It all drains away. We have to buy water in the summer and have it stored in the well. We buy 1000 gallons at a time. The water drains out at about 150 gallons a day after subtracting the amount of water that we use. We don't know why we have all the water we can use in the winter and spring, yet have none the rest of the year. Where is it going? If we have a leak in the pipe to the house, wouldn't it leak all year? This is a very difficult area to get water. We have considered trying to put a drilled well in, but have been told we probably wouldn't get water. We have considered putting in a new bored well, but what if it does the very same thing? If we don't have water, we will have to move. Can you, please, shed any light on this situation? Thanks!

PS. My husband and I were just talking and we are wondering if trees could empty the well. It makes seasonal sense to us and the well is in a densely wooded area of maple trees. The woods are a large reason that we bought this property. Thanks again.