I'm supposing you are making/building your new (double?) doors & frame.

Yes. Exactly the same as what it has now. (Sizes below)

Any good quality hinge is recommended. You would need to gauge,
for yourself, the weight of the doors, as to how many hinges would be best.
Maybe, put two on, then if you find a third is necessary, add an additional, later.

Currently it has three hinges per door of a lesser quality/strength
than the gate hinge link I posted.

As for as water entering the shed, build your framing as you would a home's entrance:
Good sound door stops, a good heavy duty sloped threshold, maybe some weather stripping,
appropriately flash/seal the installation of the frame to the rough wall, etc. Caulk well under
the threshold. Since, I suppose, you are removing the old framing/threshold (maybe exposing
the floor framing, under neath), make sure the floor joists/supports are up to par, solid, etc..
i.e., do a good inspection of the supports and flooring, while you're at it. Has any of that gotte
wet, in the past, and may be needing some maintenance or additional support?
Sonny

Not water into the shed, water pooling on the shed doors themselves.
On the front cross pieces; decorative barn style ... typical 1x4 trim
around entirety of both doors and one "X" on lower part of each door.
The trim was never caulked and so the water runs along the panels and
behind the trim, causing the trim and doors, to rot.

I'm not ripping up any flooring, or tearing apart the door framing. No
need as all is in good shape. This is a standard kit shed, 10'x16',
and it came with all the trimmings ninteen years ago.

Door space is are 5'11" wide and 7'1" tall. So 5'11"/2 gives 71"
total, roughly 35.5" per door width. I figure, with better hinges and
hasp, new wood panels and a few 2x4's, total cost will be about $100.

I want to replace the doors with minimal cost, within reason of
keeping it around at least another ten years. Otherwise I can
rent-to-own a new, bigger and better one in that time frame and not
have any hands-on work to do myself. It includes a metal roof,
windows, more vents, etc., all dropped on site pre-built.

Ultimately that would cost 5-6 times my repair cost but I already have
three people who would buy my current shed and I would not would not
have to do any work myself. I know it sounds lazy but I have been
having major back, neck and hand issues this last year which are
getting worse and I don't know how much longer I can continue to do
jobs myself.

`Casper