How can I get rid of moldy, damp soil?

pressure wash. or scrub with brush and water hose, then replace gutters.

Until you get new gutters, you might want to fill in the low spots with sand or soil so the water runs away from the house. I'd use a 'natural' addition to get rid of the yuck by mixing this combination in a bucket: 1 cup each Epsom salts, ammonia and mouthwash (cheap), and 1/2 cup cheap dish soap. Mix well to dissolve the Epsom salts and put in a hose end sprayer and give the area a good soak. This is actually a good basic spray for the soil and it won't harm plants or animals.

Keep in mind that your main issue here is water pooling; you need to get it away from your building so it doesn't cause your building's wooden structural parts to rot. Best thing to do is to take some pvc pipe & drill holes every couple inches along the length of the pipe in the TOP 1/3rd; next dig a shallow 6 to 8 inch deep trench AWAY FROM the building into an area away from the building[s] & towards or into a flower bed that will utilize the water...lay the pipe into the trench & cover it with some small rock & or sand, then with the top soil you removed. The water should drain away from the building where it had been pooling & into the area where it is needed. You can plant moisture loving plants in the drainage area to use up the water. As the water is relocated, the area that WAS damp, wet, mushy, will drain & go back to 'normal'...but remember that as long as you don't have the gutters to carry the water away...you will still have water issues. NOTE: add additional pipe lengths to reach to area you have in mind for the water to go to. ALSO, you can bury a 5 or 6 gallon bucket in the ground as a 'catch basin' at the end of the pipe...make sure that you drill several holes in the bucket bottom so the water will seep back into the ground; drill a hole in the side of the bucket for the pipe to enter the bucket & secure the lid on top.

Your first priority should be to replace the gutters. Until you do, the problem will grow beyond just having moldy soil. You will eventually have some very expensive structural damage both to the eaves (overhang) and the foundation. I encourage you to address the gutters ASAP.

look up French drain on google or youtube....a lot of work, but well worth it...and get the gutters replaced ASAP or have foundation problems and termite problems and rotten wood where the walls meet the foundation...most likely your original gutters fell because you had not cleaned them and the debris was too heavy...gutters should be flushed and cleaned at least annually...in the northern states more frequently because of all the leaves building up...no need for a ladder if you bond pieces of 3/4 inch pvc together with a curve on one end like a shepherd staff... a male coupling that fits the garden hose nozzle...on the other end a coupling to attach the hose