This isn't awkward - a lot of people have the same problem! I'm larger in the chest, too, and I would have never dreamed of wearing a dress like this back when. Lately I've found a few solutions, though.
These days it's ok to wear a tank top under a spaghetti strap dress (as long as it's casual), or even a t-shirt. Since I don't like having too many layers because of the added bulk, my solution is a sports bra. The straps look like a tank top, it gives a TON of support, and it looks fine under casual strapless or spaghetti strap dresses. I wear the Champion Jogbra (see picture below), because it comes in small back sizes and large cup sizes (which is what I need). This bra also has adjustable straps, which is great for gals who are short like me. I have to raise the straps pretty high up to keep the girls where they should be.
You will notice I wear strapless dresses every once in a while, but I usually have to either wear a shrug to hide the straps, or wear a bra where the straps look like an extension of the dress (this only works with demi bras). In this first picture, I'm wearing a bra with straps, but the straps matched the dress, and I kept my hair covering the straps. Later I wore a sweater so I didn't have to keep my hair in front the whole day.
For this dress, I wore a black bra. The straps looked like part of the dress - my husband didn't even know until I told him later on (if you look closely, you can see the strap poking out from under my hair).
For this last dress, I just wore a regular bra, and added the crocheted shrug.
If you find a strapless or spaghetti strap dress that looks great on you and the only drawback is the bra strap - there are always ways to fix that. Wear a tank underneath, get clear straps, or just add a scarf, shrug, or sweater. Nevertheless, it is hard to find the perfect spaghetti strapped or strapless dress for big busted gals, so unless you find a dress that works for you, just avoid the style. It's never worth it to "make-it-work" only for it not to be flattering on you.
