We know that using Chinese Remainder Theorem, CRT, for solving a system of linear congruences will give a unique solution $x \pmod {n_1 n_2\cdots n_t}$ where ${n_1,n_2,\ldots,n_t}$ are coprimes. But the solution $x$ is not necessary to be in the least non-negative residue form i.e $x\in\mathbb Z_{n_1 n_2\cdots n_t}$.
Is it possible to use CRT so that $x\in\mathbb Z_{n_1 n_2\cdots n_t}$? And we don't have to compute $x \pmod {n_1 n_2\cdots n_t}$.
No, CRT only states that the solution exists. To find out the explicit solution, you will have to use extended Euclidean algorithm for the case $t=2$, and then rolling the result over, in a similar manner with mathematical induction.