You seem to miss your surgeon's point, and are asking reasonable questions. Your dissolving stitches use your own body's moisture to dissolve, not external moisture. Once your incisions are sealed (usually by 3-4 days if everything is healing properly), external moisture doesn't get to the sutures unless you soak too long or scrub too hard and tear open tiny areas of your incision (both bad).
Showering is good because it cleans the wound surfaces, minimizing the risk of superficial infection which could cause wound breakdown or stitch abscess, as long as the shower is not too long, too hot, or too vigorous!
Bathing in a tub is absolutely the worst thing to do for a healing incision, especially if the incisions are immersed in the water you are sitting in. Remember what body parts are contaminated by fecal and vaginal bacteria, what parts are in the water, and which of those bacteria you want on your incisions. (This is also why some women who bathe excessively have more frequent urinary tract infections.)
Sponge bathing is a tad better, but not if you sit in the water you absorb with your sponge and use on your breasts! Can you say "Poop water?" Sorry to be so graphic, but what you really need to hear! Sponge bathing from running water in a clean sink is actually OK, but unnecessary.
Gentle tepid and short showers will cleanse you and your incisions without worry of sutures dissolving prematurely. Clean water is all that touches your breasts, and once contaminated, it goes down the drain! Using good hygiene (hand washing before touching your incisions) will avoid most risks of infection. Follow your doctor's advice and call if you have increased redness, swelling, pain or purulent (pus) drainage. Fever and chills come later, and mean you really did miss the other signs of infection, which come first. Best wishes!