Drainage

I’ve inherited quite a mature yet overgrown garden which is north facing.

We’ve had to remove a few very large trees but very conscious it would have helped with drainage and moisture in the grass. The lawn is already soft and mossy which I assume is a combination of north facing and generally wet conditions (were on north east coast)

Is there anything I can do to overall help water absorption in the garden to compensate for loss of trees. To make things worse, i need to build a shed which will also cover more area that could have helped drain. I’ve got young kids so lawn will be constantly used and don’t want it churned up due to wet conditions.

Id looked at soakaway but seems you need to have pipes to drain away into it? Is there anyway I can just dig up the garden and infill with rocks/gravel then returf to increase the volume that can hold water underneath?