I think they must be adding slots fairly frequently. (But they might also be snatched up quickly.) My wife and I put in an Amazon Fresh order yesterday after doing a quick "dry run checkout" to confirm there was an available slot. While shopping, we got a notice that the slots were exhausted, but we continued in the interest of getting an order prepared. When we checked out, it again said there were no slots. I waited a minute and refreshed the browser, and a slot had opened up for tomorrow. I'm in Colorado, but I assume AF works the same everywhere.

So far, I've had a better experience with Amazon Fresh than with Instacart (King Soopers/Kroger) due to 1) the ability to "call dibs" on items instead of waiting for the shopper to see there's no stock and struggle to find a replacement, and 2) the quick 1-2 day delivery (but maybe I've just gotten lucky). On the other hand, lots of stuff is out of stock which makes shopping challenging.

We also put in an Amazon Pantry order yesterday. Expected delivery: April 26! :O

Dozens of times per day. It's on my bookmark bar, and I click it routinely: https://www.amazon.com/afx/slotselection/

My wife placed our first Amazon Fresh order last week and received it the next day, as expected. Worked great - on time, well-packed, and it was nice to be able to see when the driver was 6 stops away.

One other option you might try - if you don't have any relatives nearby that can help, community groups on Facebook and church groups in your area may be delivering groceries to residents in need. Get on a couple of neighborhood/church mailing lists and see what's available.