Quite an unusual starting problem

So our ford taurus has had a weird problem that only got worse once we replaced the starter and gave it a fresh battery. Whenever you turn the key in the ignition, the anti-theft light will flash once, and nothing happens. Instrument cluster wont come on, nothing. Occasionally the car will start right up like it should, but its happening less often.

The way we get around it is by turning the key back to acc power, and then turning it all the way forward to starting position. Sometimes it starts up after doing it once that way, but i've also sat there doing it for five minutes or so with no luck. (Frustratingly one time i tried for a good five minutes, only to try and start it normally without thinking, and it fired right up).

The car has good power, and lock/unlocks fine from the remote. ACC power works just fine, and when started it runs and drives no problem. Even the fuses are all good.

Again, the only way it starts now is by turning the key all the way back to ACC power, and then turning it 'just right' into the starting position, repeatedly, till it starts. Try a few times, stop, try again, etc.

From what i've read, it sounds like i have a bad wire somewhere in the harness, but to be honest i have no clue why it does this.