Haunted grammar

At least two grammatical facts I know--or think I know--are so closely associated with the teachers I learned them from that I actually see these women's faces whenever I use those constructs. Using the possessive with a participle (as in "We should discuss Sally's wanting to quit her job," as opposed to "...Sally wanting to quit...") is Mrs. Gretar from high school. Saying "the lady who..." as opposed to "The lady that..." is Miss Seckel, fourth grade. ("Lady" for "woman" was just fine, but "that" implies an inanimate object. I remember sensing at the time that this was a personal issue for Miss Seckel.) Frankly I'm not sure either of these usages is correct; I find myself wondering about them, and that's when one or the other of these faces appears before me. And I get scared of trying anything else.