Tulane is not a safety unless you get in early action with financial aid at which point it becomes safety.
One of your reach schools is your first choice. I suspect that you will apply early to at least one of them. If your first choice does not restrict you from applying Early Action to Tulane then you might have a safety before final applications are due for RD and could relax knowing that you will get in somewhere. I don’t believe that Tulane requires an additional essay, so it’s just a matter of filling out a trivial application.
If however, the reach school you apply early to, such as Harvard, does single-choice, early action, then you would not be allowed to exercise this option.
I also think Lawrence is an excellent option, and it’s very likely to admit you. I don’t think they even filled up this year and were on the list of colleges seeking applications in May. They also offer early action if you can do it.