You appear to have a well with a marginal refill rate (how fast water comes into the well) and insufficient storage (water in the well above the intake level) to meet the rapid/large water demand of the washer.
So, when the washer calls for a lot of water all at once, the well is being pumped down below the pump intake level, and then the pump shuts down (at least someone put that protection on it - otherwise it will pump dry and burn out.)
There are many possible solutions which depend on the details of your well.
Sometimes you can just set the intake deeper, if the well is deeper than where the intake is set. However, if you have a surface-mounted pump, there are limits to that approach - theoretical limit is ~32 feet, practical limit is more like 25-27 feet for sucking water up.
If there's space below, a submersible pump can be set lower in the well, since it only sucks from where it is, and pushes water up. You can of course drill further to make more space below, but that's very expensive, typically.
Then there's the "blended approach" of jet pumps, which I consider an inefficient holdover from the past when electric motors were not very reliable that should be replaced with a submersible in the current era. Since you only mention having to fuss with the pressure switch we'll hope that does not apply here.
You can put a flow-limiter (aka flow-restrictor) on the pump output to prevent the pump from overdrawing the well by limiting its rate of output. You can combine that approach with a larger, or additional, pressure tank so you have more stored pressurized water to draw from. If you just use more storage without a flow limiter you may see the pump trip off more often.
You could also put a flow limiter just on the line to the washer, if that's the only use that causes a problem, as described. That's a more sophisticated version of what your clogged screen was doing crudely. A possible stopgap that requires only changing controls, no plumbing, would to try doing the smallest load size your washer can do to see if that can be supported by the well as is.