  The rich absorption spectra of Sun-like stars are enticing probes for
variations in the fine-structure constant, $\alpha$, which gauges the strength
of electromagnetism. While individual line wavelengths are sensitive to
$\alpha$, they are also sensitive to physical processes in the stellar
atmospheres, which has precluded their use so far. Here we demonstrate a new,
differential approach using solar twins: velocity separations between close
pairs of transitions are compared across stars with very similar physical
properties, strongly suppressing astrophysical and instrumental systematic
errors. We utilise 423 archival exposures of 18 solar twins from the
High-Accuracy Radial velocity Planetary Searcher (HARPS), in which calibration
errors can be reduced to $\lesssim$3 m/s. For stars with $\approx$10 high
signal-to-noise ratio spectra ($\ge$200 per pixel), velocity separations
between pairs are measured with $\approx$10 m/s statistical precision. A
companion paper assesses a range of systematic error sources using 130 stars,
with a greater range of stellar parameters, providing accurate corrections for
astrophysical effects and a residual, intrinsic star-to-star scatter of 0-13
m/s. Within these uncertainties, we find no evidence for velocity separation
differences in 17 transition pairs between solar twins. In a second companion
paper, this is found to limit local ($\lesssim$50 pc) variations in $\alpha$ to
$\approx$50 parts per billion, $\sim$2 orders of magnitude less than other
Galactic constraints.
