I don't know where you are located, but in MA where I live it is illegal to connect greywater (washing machine) to a drywell, it needs to go to the septic system. There are many reasons for this, not the least is that if you have a potable water well, you may be contaminating it with washing machine water that is flowing from your drywell into the ground, then into your well. Bad news.
It is certainly possible that the drywell has been plugged by washing machine water, which can contain soap, hair, fabric, dirt etc., none of which should be entering a drywell. You can dig another drywell further from your house, then connect the pump to that. Ideally you would like the drywell as far as possible from the foundation, this would minimize the probability of water flowing back into your basement (the short circuiting problem).
It is certainly possible that the drywell has been plugged by washing machine water, which can contain soap, hair, fabric, dirt etc., none of which should be entering a drywell. You can dig another drywell further from your house, then connect the pump to that. Ideally you would like the drywell as far as possible from the foundation, this would minimize the probability of water flowing back into your basement (the short circuiting problem).