At my previous house I had the same issue with the malformed carrots. I ended up building a raised bed with soil I brought in and that solved the problem. You could bring in some sand and well composted manure and till it into a patch to plant your carrots in. I also don't plant carrots in a row, I sow a patch 2-3' wide. I do the same for lettuce, beets, turnip and a few others. Corn, beans and peas are about all I plant in rows.