Size 13 is a juniors size, and that means that it's meant for a shorter body frame (a junior fit model is on average 5'5"). A missy fit model is taller (average 5'8"). Additionally, juniors clothes are meant for a smaller bustline, whereas missy clothes are meant for a bigger bustline. Plus, another thing is that as we age, gravity affects where our bust is postioned on the body (the older we get, the lower they get).
Changing the straps is one of the easiest alterations to do. If you're an intermediate sewer, you could do it yourself, but if you're unsure about sewing, any alteration shop can do it for you. Check the inside construction of the dress, as sometimes they leave extra length of the strap sewn on the inside (so that you can let out the straps). If they do have extra length sewn in, then all you have to do is release the amount you need and sew the straps in place. If the straps are "spaghetti straps" (made of the same fabric as the garment, but the fabric is sewn on the bias), this would be complicated to reproduce (as it takes a lot of extra fabric to make them) and you'd be better off using ribbon (or equivalent) to replace them. If there is no extra length, then you'll need to go to a trimmings store (place that sells ribbons, sequins, belting, elastic, etc.--such as M&J Trimmings in NYC) or a fabric store and buy some ribbon that matches (or compliments) the dress. There is some "stretch ribbon" which has elastic yarn in them, and those would probably be the most comfortable, but it's hard to find.