Data:
They've been traveling for 3 weeks, but not necessarily across the lake that entire time. Assuming walking speed in a straight direction 10 hours a day, that's 30km a day, for 630 km so far.
They are headed way out west, to a location with many rockets.
The lake was drained for irrigation, THEN for land for farming, in that order, for those reasons.
The lake was really huge.
The area they left was green and forested, and apparently that way any day trippable distance (read: not riverine greenness in an arid or semiarid area).
The area they are now is grassland. Not sure what's up with the exclusively dead trees.
A lot of names and stuff more closely associated with the US than other regions.
The climate stuff is not necessarily indicative, considering how much the world changed, though if this was Earth I would expect most weather systems to remain basically the same. And of course, this all depends on the whims and knowledge of Jeph, and the location of things remaining basically the same as it is today.
Having said that, the biggest indicator is that specific sequence and reasoning for draining of the lake, which is not very common. Many have been drained for land (sometimes for settlement, sometimes for agriculture), many for irrigation, but that sequence only applies to one large lake I know of; Tulare Lake. Formerly the largest lake west of the Mississippi, now nonexistent, drained first for irrigating the developing farmland in the California Central Valley, then to itself be used for land. Vandenberg, a major rocket launch facility, is reasonably southwest (well, more the south than the west) from there.
That lake would probably reconstitute itself without agriculture, but so would most drained lakes when human water usage stopped, since the ground level is still lake friendly. It temporarily reconstitutes to this day when there's enough rain and the canals can't drain it off fast enough. Also, unless they had also been heading much more south than west, there aren't a lot of forest friendly places east of Tulare Lake like we saw. Still, the area would be arid grassland if it wasn't for farming, so it's the most likely option with current data. In my opinion.