As I recall from my '66 Chrysler service manual, the adjustment on the kickdown linkage goes something like this.
With the carb at base hot idle, adjust the slotted end of the linkage (after removing any retention devices from the carb throttle linkage stud) to just touch the stud itself. THEN add two full turns of preload (lengthen the adjustment) and then reinstall.
IF this just started, then the linkage is probably fine as is. Just how "late" is the shift into 3rd gear? To gain a little better acceleration, I added 2 full turns MORE preload to the '66 Newport and that raised the part throttle shift points a few mph, which allowed more acceleration for a given throttle input, in normal driving. Min-throttle upshift into 3rd was right at 30mph.
UNLESS the trans is slipping, NO NEED to even touch the band adjustment. The old-line Chrysler mechanic at the local dealership, when I asked him the same question years ago, told me that. He said he'd tried running them "loose" and "tight", with no difference in how things acted. So, if it's not slipping, don't touch it. IF it's slipping, it needs to be replaced. But I suspect it's not or you would have mentioned that possibility.
Do a fluid/filter change and use quality atf (Dexron III equivalent as that's what Chrysler superceeded the old Type A, Suffix A trans fluid spec to in '68), or you can use the Ford Type F-spec atf, too. But the Dexron III equivalent works just fine. I'm not sure where Chrysler's atf specs or OR which one might work in the older 727s. The newer fluids are so "friction modified" to work with lock-up converters (although the Dexron III would do this too), but the newer vehicles modulate the converter lock-up in ways we didn't imagine back then.
What other automatic trans have you been around? Just curious.
CBODY67