Infinite recognizable not decidable subsets
Let $K$ be the set of indices of Turing machines which halt on the empty input. Consider the following language:
$$X = \{ 0 w : |w| \in K \} \cup \{ 1 w : |w| \notin K \}. $$
You can check that neither $X$ nor its complement are recognizable.
Therefore your proof idea doesn't work. Here is a different idea. Let $w_1,w_2,w_3,\ldots$ be the words in $L$, enumerated according length and then lexicographically (so if $L = \Sigma^*$, the order would be $\epsilon,0,1,00,01,10,11,\ldots$). The language
$$ Y = \{ w_i : i \in K \} $$
is recognizable but not decidable.