When in doubt, add compost!
If you have more patience than upper-body strength, maybe the neglected garden beds could be resurrected with a few years of leaving a cover crop on them, and using the green growth to mulch or make compost for your shed-bed. "Lasagna-style" gardeners would lay down some cardboard first, to help suppress grass and weeds, and cover the cardboard with compost.
I found mixed advice online about coal possibly contaminating soil. I didn't find any source that seemed really knowledgeable, except for some people who said it hadn't killed THEM yet.
But one of those people said she had a "lots of pieces" from lead batteries in her soil , so she put her garden "where there were fewer pieces" of lead. I guess the lead hasn't KILLED her yet, but ask people in Flint MI what they think about that!
MAYBE some organization would do free soil tests when testing only for contamination.
But if you plan to sell your yard anytime soon, bear in mind that all states I've lived in REQUIRE that you disclose any such thing that you know to all potential buyers. Results like that might be almost trivial, or far below federal standards for toxicity, but it's a conversation you might not want to have with buyers who only hear the word "lead" or "arsenic" , and don't make a big distinction between "parts per million" and "parts per trillion".
