Since no one has responded, I'll tell you what I think as a homeowner (not a pro of any kind), and my response will help bump up your thread, too.

Water needs to flow downhill, and water shouldn't be sitting damp near your foundation. (nor around your fence, imo).

I'm an amateur here, but I think you need in-ground drains that are sloped towards wherever your water needs to drain toward. In my tract neighborhood (SoCal), in ground drains throughout the backyard, drain into underground pvc pipes, that go out through the front yard, through the curb, into the street gutters. I don't know where you live or how ground water is handled where you are.

With those issues I see in your photos, I'd call my gardener, who could install drainage pipes and drains in the ground in a day or two, maybe with one guy to help him. I would also ask him to slope the ground away from the house for good measure. My gardener routinely does that kind of work.

On our house, every gutter downspout lets water out into a ground drain. I dont know why you don't have in-ground drains, but again, I'm not an expert. There are people who do only gutters, and you can check your Nextdoor website/app to find someone to adjust the downspouts, once you have proper drains.

I see a sprinkler head in your lawn - Top Picture.

Turn those sprinklers off and see if things improve!