According to Jeff Wisnia :
I think your best bet is to bite the bullet and have a new hot water
heater installed, given the age of your unit.

I think so too.

This is my reasoning: having both the water overheat _and_ the breaker
trip is suggestive that the OP has multiple faults in the heater, and
short of having an expert diagnose things, the OP may get himself
into real jackpot. Plumbing prices being what they are, hiring a plumber
to diagnose and repair the heater will probably cost nearly as much as a new
heater.

1) if the water overheated, the thermostat isn't tripping. So, you have
fairly good evidence that the T-stat is defective.
[At least you noticed this. In some cases (stuck OP valve (they get stuck
with age) and a non-return valve in the supply), a failed thermostat can lead
to a steam explosion.]

2) If the breaker trips, either something is wrong with an element, or the
thermostat is broken to the point of energizing both elements simultaneously,
or the breaker has gone defective too.

3) If both elements energize simultaneously, I would _expect_ the breaker
to trip faster than 5 minutes. Unless the double element current is only
a bit higher than the breaker.

4) It's fairly unusual for an element to draw more current than it should.
They usually simply go "open". For an element to draw more current, I
would immediately suspect that the element coatings have delaminated, and
either the upper element has broken and fallen on the lower, or, something else
has fallen on the lower element (eg: a corroded-thru anode).

I personally would expend effort to diagnose the problem[s] before deciding
to junk the heater. I hate replacing things when I'm not exactly sure what's
wrong.

In your case, I'd suggest:

Energize the unit again, and test for voltage on _both_ elements, using
a voltmeter or neon tester. If both elements are energized simultaneously,
you've identified the cause of the breaker trip - you already know the
thermostat is defective (because of the overheat), this rules out the elements
as a cause of the breaker trip.

If both elements energize simultaneously, replace the thermostat.

If the thing is still flakey, get a new tank, and return the T-stat.
--
Chris Lewis, Una confibula non set est
It's not just anyone who gets a Starship Cruiser class named after them.