The incidence of multiplicity in cool, luminous massive stars is relatively unknown compared to their hotter counterparts. Here we present radial velocity (RV) measurements and investigate the multiplicity properties of red supergiants (RSGs) in the 30~Doradus region of the Large Magellanic Cloud. We provide absolute RV measurements for our sample and estimate line-of-sight velocities for the Hodge 301 and SL 639 clusters, which agree well with those of hot stars in the same clusters. By combining results for the RSGs with those for nearby B-type stars, we estimate systemic velocities and velocity dispersions for the two clusters, obtaining estimates for their dynamical masses of $log (M_{rm dyn}/M_{odot})=$3.8$pm$0.3 for Hodge 301, and an upper limit of $log (M_{rm dyn}/M_{odot})$<3.1$pm$0.8 for SL 639, assuming Virial equilibrium. Analysis of the multi-epoch data reveals one RV-variable, potential binary candidate (VFTS744), which is likely a semi-regular variable asymptotic giant branch star. We estimate an upper limit on the observed binary fraction for our sample of 0.3, where we are sensitive to maximum periods for individual objects in the range of 1 to 10 000 days and mass-ratios above 0.3 depending on the data quality. From simulations of the RV measurements from binary systems given the current data we conclude that systems within the parameter range q>0.3, $log$P[days]<3.5, would be detected by our variability criteria, at the 90% confidence level. The intrinsic binary fraction, accounting for observational biases, is estimated using simulations of binary systems with an empirically defined distribution of parameters where orbital periods are uniformly distributed in the 3.3<$log$P[days]<4.3 range. A range of intrinsic binary fractions are considered; a binary fraction of 0.3 is found to best reproduce the observed data. [Abridged]
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