I cannot tell you when your particular wound can get wet, and I want to be plain about why: it depends on the dressing used, whether the wound has fully sealed, and your surgeon's own protocol, which is why you have been given the general instruction rather than a date. Ringing the ward, as suggested above, is genuinely the right move and not a nuisance call.
What I can give you is the general shape. Surgical wounds are usually kept dry until they have sealed, and many units now use a waterproof dressing that allows showering sooner, so the answers people get legitimately differ. Soaking, meaning baths, swimming and hot tubs, waits considerably longer than showering in every protocol I know. On the sling, the common instruction is that it may come off for washing and dressing while the arm hangs passively at your side, with no active lifting or reaching, and it goes straight back on afterwards. The sling is protecting the repair from active use, not from the air. Again, confirm that against your own instructions, because an anatomic total protecting a healing rotator cuff is often handled more strictly than a reverse.
The part that is universal and worth more than any showering tip: know what you are watching for. Spreading redness, increasing swelling, a wound that is weeping or opening, a fever, or pain that climbs day on day are all reasons to contact your surgical team the same day rather than waiting for your next appointment. Infection affects roughly 1 in 100 primary shoulder replacements, so it is uncommon, but it is the reason wound instructions are as fussy as they are. Our page on infection after a shoulder replacement sets out the signs and the timescales, and the practical side of the early weeks is in the week by week recovery account.