Is it ve'eschanan or va'eschanan?

This is a fairly straightforward question to resolve. Just look at the pasuk in a Chumash, and it is:

כג וָאֶתְחַנַּן, אֶל-ה', בָּעֵת הַהִוא, לֵאמֹר.

23 And I besought the LORD at that time, saying:

and there is no dispute on this matter in masorah. But in popular speech, it often becomes ve'eschanan, with a sheva under the vav rather than a kametz. Thus, a Google search yields the following number of hits:

ve'eschanan: 1,530 hits

v'eschanan: 4,140

vieschanan: 76

vietchanan: 2

v'etchanan: 11,700

ve'etchanan: 14,200

In contrast, with a kametz can also be much more popular:

va'etchanan: 134,000

va'eschanan: 23,000

vaeschanan: 37,400

and so on. I would guess that you can probably find some interesting demographic differences between these two clusters.

Why is it vaetchanan, with a kamatz under the vav? This is the difference between vav hachibbur and vav hahippuch. If it would be vav hachubbur here, there would be a sheva under the vav and the intent would merely be "and", but not to reverse the apparent imperfect (future) tense to perfect (past) tense.

A vav hahipuch, on the other hand, would take etchanan, which is a first person imperfect, and change it into a perfect (past) tense word. And this is what we want, since Moshe is talking about past rather than future actions.

For a typical vav hahipuch, there is a patach under the vav and a dagesh in the next letter. This is not possible here, for the aleph is a guttural letter and thus does not take a dagesh. Instead, we have compensatory lengthening, in which the patach in lengthened into a kamatz. The weirdness of this form, combined with the softness with which we Ashkenazim pronounce the kamatz, is quite likely behind this popular mistaken pronunciation of it as if with a sheva.