Ok. So one light goes from blinking to solid just fine, and the other one is just solid, no matter the switch position. Basically, this means that the non-working LED always has ground connected to it, regardless of the output of the CD4047BE (which we know is working properly, due to the other LED). Therefore, it could be:
1) Both pins 4 and 6 on the switch are connected to ground, or they are somehow accidentally shorted to each other (referencing the switch pinout I posted earlier).
2) You have your source pin and your drain pin flip-flopped on that mosfet. Be sure the pinout matches up with the circuit diagram.
3) Your drain pin and source pin on that mosfet are accidentally shorted to each other. Check to be sure there isn't any stray solder making a connection between the two.
4) Your mosfet is blown and is shorting out (I really doubt that this is the case). To check this, with the circuit powered down (no battery), measure the resistance between the source and the drain. If it is <10 ohms, then the mosfet is bad.
That's about all I can think of. Post again if none of these do the trick, and I'll try to come up with something else. Good luck!