We modify your approach by first placing the A's, then placing the I's.
We can arrange the six distinct letters C,L,F,O,R,N in $6!$ ways. This creates seven spaces, five between successive letters and two at the ends of the row.
Case 1: We choose two of these seven spaces in which to place the two A's, thereby separating them.
We now have eight letters. This creates nine spaces, seven between successive letters and two at the ends of the row. We choose two of these nine spaces for the I's. The number of such arrangements is
$$6!\binom{7}{2}\binom{9}{2} = 544,320$$
Case 2: We place both A's in the same space.
We again have eight letters. This again creates nine spaces. The space between the two A's must be filled with an I. Therefore, there are eight ways to choose the position of the other I. The number of such arrangements is
$$6!\binom{7}{1}\binom{8}{1} = 40,320$$
Total: These two cases are mutually exclusive. Hence, the total number of arrangements of the letters of the word CALIFORNIA in which no two consecutive letters are the same is
$$6!\left[\binom{7}{2}\binom{9}{2} + \binom{7}{1}\binom{8}{1}\right] = 584,640$$
which agrees with the result obtained by using the Inclusion-Exclusion Principle.
We shall solve by successively applying the well known "gap" and "subtraction" methods.
Firstly, we shall keep the $E's$ separate by placing them in the gaps of $-A-B-C-D-D-$ and permute the other letters, thus $\binom63\cdot\frac{5!}{2!} = 1200$ ways.
We shall now subtract arrangements with the $D's$ together treating them as a super $D$, $-A-B-C-\mathscr D-$, i.e. $\binom53\cdot4!= 240$
Thus we get $1200-240 = \boxed{960}$
Best Answer
This is an alternative approach to compute this numbers using a computer. The set of all words we are trying to count is of course finite, to it is a regular language. Luckily, it is easy to construct an automaton which recognizes it, so we can look at the adjacency matrix of the machine and compute its powers. The obvious automaton has $(2+n)n!$ vertices, which is pretty huge.
The actual computation of the matrix powers takes time similar to $$\log\binom{n+1}{2}(n+2)^3n!^3$$ if one uses repeated squaring.
In mathematica, we can code this with
Using this code, I can evaluate
to get
in 57 seconds.