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I fold my foil over the lip of the drip tray also…in spite of the recommendation not to. Supposedly, heat/smoke gets up under it when it goes over the lip.Two things I have found to help eliminate grease getting under the foil. First, check the foil you are using for pinholes. On occasion, I have found recurring pinholes that appear to have occurred during the manufacturing process. When I find this situation, I use two layers of foil, making sure that the pinholes are significantly separated from one another.
Second, I fold the edges of the foil about 1/4” over the lip of the drip tray edge so that any grease that happens to drop there will not run down the exposed edge and under the foil. I have seen cautions about running the foil over the edge of the drip tray, but the slight 1/4” overlap doesn’t seem to cause any issues.
As for washing the drip tray, the one for my 340 is small enough th fit in the sink, standing on end. It will also fit in the dishwasher. If I had one that wouldn’t fit in the sink, I think I would consider getting a large-size “hotel pan” that the drip tray would fit in, standing up, and that I could set on the counter. A bit of care would be needed to avoid a mess, but I think it would work. YMMV