a third or fourth opinion is always a good idea so go for it. Both
contractors are correct: patches never look as good or last as long as
done resurface (look at asphalt patches in the road and
consider how well they work), however you MIGHT be able to get away
with a patch if it's in a low traffic area and it's kept well sealed.
Ask the contractors if they warranty their work - will they come back
and fix it next year if it fails, how about two years, etc?
Two things are working against you:
1) the sinking is probably caused by an poor base under your driveway.
Not enough or incorrect type or incorrectly compacted fill, maybe not a
proper binder layer. You can keep filling the sinking spots, but
they'll continue to cause problems. The only true answer to this that
I've ever heard of is to actually tear out the driveway and start over.
This will probably cost a lot more than you want to consider. Perhaps
the contractors know of a way to fix this that I haven't heard of, but
I'd be very careful about it (see above comments about warrantying
2) cracks allow water in and under the asphalt, which expands and
contracts during the freeze/thaw cycles, which widens the cracks and
causes more cracks and moves the driveway surface up and down. The
shaded part of your driveway doesn't thaw as often as the sunny part
during the winter, so this cycle works more slowly on it. It too will
show damage over time from this, but it'll take longer. Keeping a
driveway sealed will help prolong it's life.
As for the boards, Trex is a plastic/sawdust composite board. It
should last longer than any untreated natural wood, but it probably
won't look quite the same. If you're ok with that, then go for it. It
will probably cost a bit more than wood too.