You're right, Ray, it does seem very odd indeed for "Plantar Fasciitis" to come on in both feet at exactly the same time.
Perhaps you have made a discovery (with the hot feet) about some previously unrecognised aspect of "PF".
Possibly there could be other explanations/diagnoses ~ so keep your mind open.
The "worse first thing in the morning, or after rest" does certainly sound suggestive . . . but let's hope it is something other than / better than PF, since that condition usually takes umpteen months to disappear entirely.
The good news is, once fully settled, you are unlikely to get it a second time around.
The bad news is that most treatments do little for the condition.
You'll hear a hundred recommendations of treatments, arch supports, and so on, that helped "me" (friends, neighbours, cousins . . . . ) ~ but when you allow for individual differences, enthusiasms, placebo effects, financial interests, mistaken diagnoses, etcetera, etcetera . . . it doesn't amount to even the small hill of beans that could impress a careful skeptic.
Not surprising that treatments are feeble, since there's very limited scientific understanding of what's going on in PF or other so-called "chronic inflammations" that have minimal actual inflammation present.
Why is it that Mother Nature doesn't just clear it up in 3 weeks, like with ordinary sprains?
More info available from Wikipedia [not too bad, really! . . . but skepticism is need about the list of treatments] and "aafp.org" which has a 2005 review of tests of treatments, and comes to the conclusion that: "No evidence strongly supports the effectiveness of any treatment".
But few of us can resist the temptation to try this or that treatment : so if I were yakking to you over the back fence (and the PF was definite) I'd say try some ibuprofen tablets at breakfast and lunch during the worst days, or when you absolutely have to do a lot of walking that day . . . and otherwise self-treat by gripping & stretching the toes upwards just 5 times twice per day (up to moderate pain level only).
The ibuprofen won't give you more than 25 - 50% pain relief; and the sole stretches (not calf stretches) will only be a mild help over the longer course.
Don't spend too much time or money on faddy treatments . . . or perhaps you can do a scientific trial of treatment on only one foot ~ and see which foot gets better faster!
Remember the old saw : "If it gets better quickly, it wasn't plantar fasciitis."