Yes, they usually do offer some sort of deal in April, for the last several years anyway. Also, if your week is Easter week, you should still see a discount (we go to WDW that week every year). There's a PIN code out for early 2017 and it does include Easter so that's always a good sign. Only problem we have is full prices are so inflated nowadays that the discount seems to hardly make a dent.
From our experience, the crowds are heavier in early April -- one of the busiest times for us was when Easter fell at the tail end of March/early April because it collided with a lot of college breaks. However, we have better luck crowd-wise when Easter is mid to late April. Plus the more into April it gets, the better the weather. It's *usually* very nice in April!
Good luck getting your deal!