I am facing a problem when thinking about the case where a unit root is present. E.g., a=a(-1)+u.

I am told that a value has to be given for a to pin down the STEADY STATE and linearization is then possible. But the problem is, a may drift far from the prespecified value, so are other endogenous variables. As a result, the linear approximation (first order approximation)around the prespecified steady state may be inaccuracy.

Is it true?

Yes, this is true as the approximation is always only local. It is typically not an issue for e.g. IRF analysis at the initially specified point, but might be a problem when doing long simulations where the variable drifts away from the initial approximation point.