How were the absolute masses of atoms calculated before the mass spectrometers?
Absolute mass can be determined electrochemically, once the coulomb is defined in terms of numbers of electrons, as through the Millikan oil-drop experiment, to be  $6.24×10^{18}$ electrons per coulomb.

See Determination of Avogadro's Number and Absolute  Determination  of  the  Electrochemical  Equivalent and  the  Atomic  Weight  of Zinc.

One thing that caused confusion in the determination of atomic mass is natural isotopic separation, e.g. sulfur derived from condensation from vulcanism vs. from mineral compounds, so that there were slightly different experimental results depending on source.