I would say the answer is contained right in both verses. Gen 7:6 says Noah was 600 "when the flood of waters was upon the earth," while 8:13 says "the waters have been dried off the earth" (emphasis added in both verses). The waters cannot be both "upon" and "dried off" at the same time. Gen 7:6 is about a year before 8:13.
Update from Comment
The Hebrew word there (היה; hayah) is in the perfect tense, so "hath been" is a perfectly fine translation, as is "was." This is because the perfect in the Hebrew has a broad range referring simply to a condition/state that is present at the time, which English would put a present tense verb (is/am), or a simple past (was/were); or it can have a concept of completed action viewed as a whole, which may or may not be actually finished yet, but simply described that way. That it could/would be described as completed by Moses would be no surprise, since the event was long past by his time.1
However, even if viewing it as a "completed action" idea, what action is 7:6 stating was completed? Not necessarily the whole period of the flood itself, but rather at what point the flood had reached a stage of having "waters upon the earth," which was only 7 days (7:10) from the time God told Noah (7:1-4) to enter the ark. The waters remained upon the earth until they were dried up, but the point is that on that 7th day from the time that God told Noah to "load up" (so to speak) the waters were "upon the earth," and so Gen 7:6, is simply a statement that sometime during the time the waters were such, including 7 days from warning, Noah was 600 years old.
As I noted in my answer to the other question, 7:6 is likely a summary statement of the flood event. However, contextual location in the text would seem to argue against it being viewed as making a statement in any way specifically tied to the exact timing of Gen 8:13. There is simply no warrant (nor need) to make that connection. All Gen 7:6 is saying is that during the full stage flood time, Noah was 600 years old. That he was still 600 at the end (being in his 601st year) may be relevant in determining a broad range of when he turned 600 during the event (since it lasted a little over a year), but other than that, there is no need to equate the two.
That all the major translations make it a reference to the fact of without reference to completion of the event ("was"/"were" in KJV/NKJV; "came" in ESV/NASB/NIV) is strong indication that the translators do not see a warrant to shift to a "hath been" emphasizing the past aspect, as much as emphasizing the "state of" the waters at the time Noah was that age.
1Information on the perfect tense in Hebrew is from Ronald J. Williams, Williams' Hebrew Syntax, 3rd. ed., p.67