That's certainly literally accurate, but I worried that it might be interpreted merely as "I didn't have both", i.e. that there was at least one which I lacked.

English doesn't bracket in the same way as mathematical expressions. In particular, verbs associate with each following noun (distributive? possibly).

Thus "Do you prefer X or Y" is not "Do you prefer (X or Y)", it's "Do you prefer X, or do you prefer Y?"

Similarly "I lack A and B" expands to "I lack A and I lack B", rather than "I lack (A and B)"