Context: The influence of binarity on the late stages of stellar evolution remains an open issue. Aims: While the first binary post-AGB stars were serendipitously discovered, the distinct characteristics of their spectral energy distribution (SED) allowed us to launch a more systematic search for binaries. We selected post-AGB objects, which exhibit a broad dust excess starting either at H or K, pointing to the presence of a gravitationally bound dusty disc in the system. We initiated an extensive multiwavelength study of those systems and here report on our radial velocity and photometric monitoring results for six stars of early F type, which are pulsators of small amplitude. Methods: To determine the radial velocity of low signal-to-noise ratio time-series data, we constructed dedicated autocorrelation masks based on high signal-to-noise ratio spectra, used in our published chemical studies. The radial velocity variations were analysed in detail to differentiate between pulsational variability and variability caused by orbital motion. When available, the photometric monitoring data were used to complement the time series of radial velocity data and to establish the nature of the pulsation. Finally, orbital minimalisation was performed to constrain the orbital elements. Results: All of the six objects are binaries with orbital periods ranging from 120 to 1800 days. Five systems have non-circular orbits. The mass functions range from 0.004 to 0.57 M&sun; and the companions are probably unevolved objects of (very) low initial mass. We argue that these binaries must have evolved through a phase of strong binary interaction when the primary was a cool supergiant. Although the origin of the circumstellar disc is not well understood, the disc is generally believed to have formed during this strong interaction phase. The eccentric orbits of these highly evolved objects remain poorly understood. In one object, the line-of-sight grazes the edge of the puffed-up inner rim of the disc. Conclusions: These results corroborate our earlier statement that evolved objects in binary stars create a Keplerian dusty circumbinary disc. With the measured orbits and mass functions, we conclude that the circumbinary discs seem to have a major impact on the evolution of a significant fraction of binary systems. based on observations collected with the Flemish 1.2 m Mercator telescope at Roque de los Muchachos (Spain), the Swiss 1.2 m Euler telescope at La Silla (Chile) and the 0.5 m and 0.75 m telescopes at SAAO (South-Africa). Tables [see full textsee full text]-[see full textsee full text] are only available in electronic form at http://www.aanda.org Postdoctoral fellow of the Fund for Scientific Research, Flanders. Ph.D. student of the Fund for Scientific Research, Flanders.