This follows from the Kronecker's approximation theorem (With the conditions modified. See edit below).
Assume without loss of generality that $a_k$ are positive, that $c_1 \leq c_k$ and $0 \leq c_k < a_k$. Let $n_1=n$ and $n_k=\lfloor na_1/a_k\rfloor$ for $k=1,\ldots,K$. Then $n_1 a_1+c_1-(n_k a_k+c_k)=n a_1 -\lfloor na_1/a_k\rfloor a_k+c_1-c_k=c_1-c_k + $ { $ n a_1/a_k $ }$a_k$
By Kronecker's approximation theorem there exists some $n$ such that $|c_1-c_k + $ {$n a_1/a_k $}$a_k|<\epsilon/2$ for each $k=1,\ldots,K$. The conclusion follows from the triangle inequality.
Edit: As Noam D. Elkies remarked in his answer what we use here is in fact the linear independence over $\mathbb Q$ of the reciprocals of the numbers, $1/a_k$ (Or equivalently in my application of the Kronecker's approximation theorem, the numbers $a_1/a_k$ ), not the numbers $a_k$ themselves. This means that the question as posed is not true, but it is true when the conditions are modified.