Scott,
You could try adding this
-"NYT cooking"
(exactly as shown) to the From: line in addition to what's already there (with a space between them).
I don't know why the thing you tried did not work. Perhaps you spelled something wrong? (I know that sounds silly -- but my keyboard has started giving me double letters sometimes which I don't always notice.) And I'm assuming that you waited for new messages to arrive, before concluding that the modified filter did not work. Changing the filter won't affect any messages already received and labeled.
Alternatively, look for something else unique to all the non-cooking messages that you can add to your non-cooking filter.
And another possible alternative if neither of these works: I think you can add
label:NYT-cooking
(or whatever is the label name you used for the cooking emails) to the "Doesn't have" filter line. Therefore, messages with that label ought to be excluded from the filter for the non-cooking emails. Gmail might complain, but it seems to work for me.
Andy