If you TT forward in time, your turnips will not rot unless you TT past 6am on the following Sunday. TTing backwards to a time before your last save, however, will make them rot. I missed the morning prices every day this week, so I TT'd back to 9am to see the morning price, and since I hadn't loaded up my game yet that day, my turnips were perfectly fine. I'd then immediately TT back to the current time and get my afternoon price, and my turnips would still be unaffected.
I should probably add that I changed the time using the clock in System Settings. I don't know how changing the in-game clock instead of the system one would affect them. But yes, TTing forward will not affect your turnips if you don't go past 6am the following Sunday.