You're about a month away from it being tank tops and mesh shorts...
As for Signatures, dress shorts with a polo shirt and sneakers should be fine. I used to pack nicer clothes: collared and chinos, but then I took a look around and realized I was overdressed. People simply don't dress up for the nice restaurants.
Now, if you wanna dress up a bit, then go for it. Be comfortable, but look presentable.
We go during Christmas week, so we usually like to dress a little nicer for our "big dinners". I'll bring darker jeans and an oxford shirt, or I'll wear a pressed t-shirt (V-Neck or something). I try to avoid bringing too many sneakers. I have my park sneakers that are comfortable and easy to walk in, and I'll usually bring a pair of boat shoes.