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Searches for variations of fundamental constants require accurate measurement errors. There are several potential sources of errors and quantifying each one accurately is essential. This paper addresses one source of uncertainty relating to measuring the fine structure constant on white dwarf surfaces. Detailed modelling of photospheric absorption lines requires knowing the underlying spectral continuum level. Here we describe the development of a fully automated, objective, and reproducible continuum estimation method, based on fitting cubic splines to carefully selected data regions. Example fits to the Hubble Space Telescope spectrum of the white dwarf G191-B2B are given. We carry out measurements of the fine structure constant using two continuum models. The results show that continuum placement variations result in small systematic shifts in the centroids of narrow photospheric absorption lines which impact significantly on fine structure constant measurements. This effect must therefore be included in the overall error budget of future measurements. Our results also suggest that continuum placement variations should be investigated in other contexts, including fine structure constant measurements in stars other than white dwarfs, quasar absorption line measurements of the fine structure constant, and quasar measurements of cosmological redshift drift.