Half-Adder Digital Logic: Not as Expected
You have misunderstood how AND gates work.  Here is a very nice list of truth tables for various digital logic circuits.  AND gates are only high when both inputs are high.  You are thinking of an XNOR gate which produces a high output whenever both inputs are the same.  The AND truth table is:

1in 2in out
0   0   0
0   1   0
1   0   0
1   1   1


and the xor is:

1in 2in out
0   0   0
0   1   1
1   0   1
1   1   0


Another thing you can do is check to make sure the AND gate is working as it should.  Disconnect the XOR gate and connect the two inputs directly to the AND inputs and make sure the outputs match the truth table.