How do I get Musescore to account for key signature while converting to tabs?

You shouldn't need to transpose to do this - if you type "B", or click on the "B" line, you should get a Bb automatically since the key is Ab. It's hard to say from just a picture what has gone wrong - much better to attach your actual score - but one obvious problem I see is that you seem to have somehow turned on the special-purpose "insert" mode, and now you have too many beats in the measure. This might also explain whey the key isn't be9ng seen correctly, depending on the exact order you are doing things in. Anyhow, turn off that special mode and go back the regular "step time" note input mode mode unless you have some unusual special reason to want extra beats in your measures.