Check out this discussion from the spring on applying ED vs. RD vs. EA.
Tulane admission paths are a lot more complicated now than they were a few years ago. Because Tulane is one of only a few schools that has all of RD, EA, ED1 and ED2.
In the past three years, Tulane has gone from filling 0% to 25% of its seats through ED. Expect that ED trend to continue going forward. Which means that the RD admit rate has plummeted.
Best guess is that the EA admit rate and the ED admit rates are both about double the overall 17% admit rate. Which means the RD admit rate is probably approaching 10%. I don’t think you want to be playing RD.
Traditionally, Tulane has given out a lot of merit money to high stat kids. These days, I think a lot of those kids (like 75th percentile stats) apply EA.
ED looks to be the preferred path for kids with 25-50th percentile stats (which is where your current SAT score falls). So if it makes sense (financially and otherwise), ED1 or ED2 seems like a good option for you.
Tulane has always been VERY big on demonstrated interest in admissions. Applying ED is the ultimate demonstration of interest. Might make sense to reach out to you regional rep and discuss (which is yet another way of demonstrating interest in Tulane).