I have used an EAP program a couple times. They are usually designed as short (4-6) weeks max therapy engagements and after that if you still need help you are referred to the behavioral side of the health care benefits.
It sounds like you are looking more for accommodations, as opposed to quick, just get you on your feet again, type of treatment.
As mentioned above, FMLA is going to be a better tool, but there is also Americans With Disabilities Act which recognizes mental health conditions as disabilities and so your employer may have legal obligations to adjust your working conditions or expectations to accommodate your disability. I have not applied for ADA accommodation but I have always been free to schedule myself. I have deliverables and as long as I have working, tested code ready to release on time, I could be hiking in the Alps for all my higher ups would care.
If I was being paid by the hour, though, I would require some accommodation because I would also face scheduling dilemmas with therapy and med appointments and other treatment related adventures.
You mentioned your new job is at a major retailer. As long as that retailer is not majority owned by people who all have Walton as their last name- then they are more likely to have a helpful/employee friendly HR team. Exceptions do exist, but mostly, the larger a corporation gets the more likely they will be eager to avoid any lawsuits from wrongfully terminated employees.
The real purpose of HR is not to help you. It’s to keep the company from getting sued. This does work in your favor though, because they are therefore motivated to stick to the legal requirements of FMLA and ADA. A smaller org might not even have an HR department, in which case you might be at the mercy of how ethical and knowledgeable your manager is about necessary accommodations.