Instead of a squeegee, try a damp but wrung out sponge. Works great. OH, it needs to be one of the cello type sponges to not leave lint and don't use it for anything else.

For the water issue, I suggest get one of the water filters for under the sink which removes "most particulate matter and hard deposits". I think they are about $20 for the contraption and about $8 for the filter and it works for about 3 months. Looks like a plastic jar with a filter, nothing fancy. Easy to install.