Circumbinary planets, orbiting both stars in a binary system, offer the opportunity to study planet formation and orbital migration in an environment different from that around single stars. However, despite the fact that $ 90~\% $ of binary systems in the solar neighbourhood are spectrally resolved double-lined binaries, there has been only one detection of a circumbinary planet orbiting a double-lined binary using the radial velocity method so far. Spectrally disentangling binary components is challenging due to blending of spectral lines and inaccuracies in spectral modelling. These inaccuracies add scatter to the measurements, which can hide the weak radial velocity signature of circumbinary exoplanets. We have obtained new high signal-to-noise, high-resolution spectra with the SOPHIE spectrograph, mounted on the 193 cm telescope at Observatoire de Haute-Provence (OHP), for six bright, double-lined binaries for which circumbinary exoplanet detection has been attempted in the past. To extract radial velocities, we use the DOLBY code, a recent method of spectral disentangling using Gaussian processes to model the time-varying components. We analyse the resulting radial velocities with a diffusive nested sampler to seek planets, and compute sensitivity limits. We do not detect any new circumbinary planet. However, we show that the combination of new data, new radial velocity extraction methods, and improved statistical methods to determine a data set's sensitivity to planets leads to an approximately one order of magnitude improvement compared to previous results. This improvement brings us into the range of known circumbinary exoplanets and paves the way for future observation campaigns targeting double-lined binaries.