Electronic – How to display 2 digit number in binary adder circuit

You have a 4-bit full adder, so far OK. Then you connected a control input to the carry-in of the lowest bit. When you keep that control at 0 you still have your 4-bit adder. When you put it at 1 you will get the addition + 1, just as you described. Putting an inverter after the first output bit will 'solve' this ONLY for additions that had a 0 in the lowest bit (go figure out why).

If you want the control = 1 to give you the addition, and control = 0 must give 'something else, does not matter what' you could put the inverter in the control line.

BTW a 4-bit added will give you a 5-bit result. If you need an 8-bit 2s-complement result look up 'sign extension'. (This smells of homework, so I won't give away everything.)

Two's-compliment overflow is not very simple when it comes to the logic involved. Have your XOR gates handy.

For an addition operation, the overflow bit should be set if the sign is the same for both operands that were being added, but different from the sign of the result (an XNOR and an XOR).

For a subtract or compare operation, the overflow bit should be set if a positive number (sign=0) is subtracted from a negative number (sign=1) and the result is positive, or if a negative number is subtracted from a positive number and the result is negative (I'll let you do that one).

For the subtract, it might be simpler to involve CARRY in the logic.