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Do law schools care how long out of college graduation you are when you apply? If you've been out for say 4 years, would they then be looking for substantial work experience? What if during those 4 years you really were just working as a paralegal, rather than anything advanced or technical, like engineering or business or whatnot? I'm asking because I'm just planning for the worst case scenario of me waiting it out a little bit more if my LSAT score is not in the range that I want it to be by the time I'm actually planning on applying, which is shorter than 4 years. I'm specifically aiming for the T6 (most ideally one of the HYS).

No. This is not something that factors into your law school admission in any way. If you’ve done something incredibly exceptional, you might see a slight admissions boost. But it won’t be held against you for being a paralegal, gardener, skydive instructor, baker or whatever else during your gap. What matters is GPA/lsat, and, less so, your admissions essays.