Here's the applicable regulation updated 1/1/12

26-112-45. Species limitations

(b) American shad (Alosa sapidissima) and Hickory Shad (Alosa Mediocris).

(1) American shad may be taken only by angling.

(2) In those streams open to fishing throughout the year the taking of American shad from July first through March thirty-first, both dates inclusive, is prohibited. In all other areas the taking of American shad from midnight on July first through 6:00 a.m. on the third Saturday in April is prohibited, except as provided in section 26-112-45(b)(4).

(3) The daily creel limit for American shad and hickory shad shall be six fish of both species in the aggregate, except as provided in section 26-112-45(b)(4).

(4) The open season for taking American shad from the Pawcatuck River, North Stonington-Stonington-Rhode Island, shall be as provided by Rhode Island regulations authorized by section 20-1-12 of the General Laws of Rhode Island and the daily creel limit for American shad in this area shall be one.

26-112-44. Closed seasons
(a) Streams. The closed season for fishing in streams shall be from midnight on the last day in February through 6:00 a.m. on the third Saturday in April, except as provided below:

(1) Streams open to fishing throughout the year. There shall be no closed season for fishing in the following streams, or portions thereof, as specified:

(B) Connecticut River and its coves.

So - Cliff Notes version - CT River is open year round, that means no Shad from July 1st through March 31st inclusive. April 1st you are good to go until 11:59:59pm on June 30th.