Certificate Definition of NL
The problem is that you won't be able to verify if the certificate is legitimate as you are making certificate with assignment of variables clause by clause. To check it is legitimate you would have to traverse back on certificate tape which you won't be able to. For example let $\phi = \{ (x_1 \vee x_2 \vee x_3) \wedge (\overline{x_1} \vee x_4 \vee x_5) $. And say your certificate is {(True,False,False),(True,False,False)}. In this case you would declare that assignment makes $\phi$ True but that's not the case as $x_1$ and $\overline{x_1}$ can't both be True and to check all assignments are valid you will have to traverse back and forth on certificate tape which you can't do. Thus your approach is not correct.