The grass looks greener but thinner and taller right? Like Rob was saying, the chemistry on the icky lawn leached off and fertilized the seeds of your grasses in the shade.
Fertilizer is not good for shaded plants that just don't need much nitrogen or phosphorous or potassium. Not enough light to get those photosynthetic factories producing at a high level.
A little too much nitrogen makes more top growth than the plant can support in the shade. Shade makes leaves wider,larger and thinner. The plants do this to be able to gather as much light as they can in the shade.
Your other grass in the sun are not healthy and that is a big problem.
You have cool season grass. Here are the rules for cool season grass crop lawns:
Mow no shorter than 3 inches. Best is 3 1/2 inches. I am not kidding!!
Water very deeply Use a shovel to slice into the soil of the lawn bed. When the moisture gets to 4" below the surface that is the amount of time to water this lawn. Not so good for your plant beds. We can discuss separating plant beds from the lawn irrigation system.
Sometimes it takes a half an hour to an hour to wet the soil of the lawn bed properly. Takes less and less in the future to get moisture down 4". 6" is best with a fully trained lawn.
You are training your grasses to make their roots grow DEEP. Cool Season Grasses have genetic large root systems. We use this in our favor.
The ONLY time you water again is when you are able to see your footprints on the grass, the grass blades stay down. This is the only time you water again!!
Watering a little every day is just what the weeds love. Watering once per week or more kills weeds. Weeds have shallow root systems. You make your crop have large deep roots to reach the moisture 6" below...the weeds on the surface DIE in between waterings.
Mow no shorter than 3 better is 3 1/2 inches. This height shades the soil so weed seeds do not germinate. Evaporation of moisture is inhibited. Cool season grasses need that top growth to feed the large system of roots. Mowing too short actually kills cool season grasses.
Aerate with core aerators leaving the plugs of soil and sod right where they fall ONCE PER YEAR.
Fertilize with the proper formulation for the time of the season; Early Spring, late Spring, Summer and Fall! 4 applications per year. I have to tell you to try Dr. Earth's Lawn Fertilizer. Costs more but only 3 applications are necessary, slow release. As an expert in cool season grass lawns, this one fertilizer blew me away. Mycorrhizae and thatch eating bacteria are included.
Where fertilizer is concerned:
Less is Best, More is Death and None is Dumb.
The moisture stays longer down in the shade, common sense, right? Grass doesn't belong in shade or even partial shade. I use gravel on the under story soils.
Stop watering every day. Water DEEPLY. Try a half hour on a zone for the lawn. See how deep that moisture goes. If it goes down 2 inches, double the time to an hour. If you have slopes and drainage you'll need to do this in 15 minute stages interspersed by the other zones watering plants.
We can talk about the timing and scheduling for your timer. The major point is that your grass needs to be watered DEEPLY (check the 4" into the soil) and then NOT WATERED AGAIN until you see your footprints on the grass, the blades will stay down!
Grass doesn't have to be everywhere...