Sounds like by manually downshifting you are screwing things up. It's possible there is something wrong with the transmission, a fluid pressure / valving leak somewhere. But I really don't think the OP is helping the situation by running Tow Mode AND screwing with the shifter.
I also have a vague recollection that you're not to use (Over)Drive if in Tow Mode. But I'm looking for references on that.
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Towing info starts on page 4-50. 4-59 has some stuff about gear selection, but not what I was looking for.
http://www.vadengmpp.com/owners-manual/chevrolet/2001-Chevrolet-Suburban.pdf
Tow/Haul Mode
Tow/Haul is a feature on automatic transmission
equipped vehicles that assists when pulling a heavy
trailer or a large or heavy load. See Tow/Haul Mode
on page 2-31 for more information.
Tow/Haul is designed to be most effective when the
vehicle and trailer combined weight is at least 75 percent
of the vehicle’s Gross Combined Weight Rating (GCWR).
See “Weight of the Trailer” later in the section. Tow/Haul
is most useful under the following driving conditions:
• When pulling a heavy trailer or a large or heavy load
through rolling terrain.
• When pulling a heavy trailer or a large or heavy load
in stop and go traffic.
• When pulling a heavy trailer or a large or heavy load
in busy parking lots where improved low speed
control of the vehicle is desired.
Operating the vehicle in Tow/Haul when lightly loaded
or with no trailer at all will not cause damage. However,
there is no benefit to the selection of Tow/Haul when the
vehicle is unloaded. Such a selection when unloaded
may result in unpleasant engine and transmission driving
characteristics and reduced fuel economy. Tow/Haul is
recommended only when pulling a heavy trailer or a large
or heavy load.
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I've towed a variety of things up and over the stiffest grade in CA (the Grapevine) fairly frequently, same drivetrain as the OP, and I've never had the problem he's describing. I have to wonder what the total weight is that the OP is dragging, he could be exceeding the GVWR, contributing to his trouble. Cherokees are about 4500# and that's a substantial trailer. I'd say he's quite likely at least 7000#, plus the Sub, plus loaded for a day/weekend at such a place, he's almost certainly exceeding the 12,000# max recommended combined weight and probably by at least a 1000-2000#.