I don't think so. First, sugar babies are one of the fastest plant-to-harvest melons that exists. The seed packets say 75 days from seed planting to harvest and if you got plants you will be a TINY bit ahead of that. Second, even if they are late, watermelons will generally continue to bloom and produce right up until the frost kills them IF they haven't produced already. In other words, once a plant/vine produces a lot of melons for a few weeks, it will decline a lot and perhaps (but certainly not always) die of what would equate to "old age". Some may disagree because it is true that many vines will bloom long after their peak, but in general even though they bloom their production and health and growth declines a great deal after they peak (say 120 days for most watermelons). But a watermelon planted quite late that hasn't yet produced will continue to bloom, put on healthy new growth, and produce new melons right up until frost kills them. Even RI is more than 75 days from your first frost (I think!) so I bet you will be ok.
On the downside, and I hate to ever rain on anyone's parade, but sugar babies aren't a great watermelon to me, in large part because the ratio of seeds to flesh is just crazy. They are probably the seediest watermelon of the 58-60 I've grown. But they're good tasting. Enjoy!
UPDATE: I'm with fruitnut all the way! You've still got time for a better melon than sugarbaby!!