Previous XMM-Newton observations of the thermally emitting isolated neutron star RX J1605.3+3249 provided a candidate for a shallow periodic signal and evidence of a fast spin down, which suggested a high dipolar magnetic field and an evolution from a magnetar. We obtained a large programme with XMM-Newton to confirm its candidate timing solution, understand the energy-dependent amplitude of the modulation, and investigate the spectral features of the source. We performed extensive high-resolution and broadband periodicity searches in the new observations, using the combined photons of the three EPIC cameras and allowing for moderate changes of pulsed fraction and the optimal energy range for detection. A deep $4sigma$ upper limit of $1.33(6)%$ for modulations in the relevant frequency range conservatively rules out the candidate period previously reported. Blind searches revealed no other periodic signal above the $1.5%$ level $(3sigma$) in any of the four new observations. While theoretical models fall short at physically describing the complex energy distribution of the source, best-fit X-ray spectral parameters are obtained for a fully or partially ionized neutron star hydrogen atmosphere model with $B=10^{13}$ G, modified by a broad Gaussian absorption line at $385pm10$ eV. The deep limits from the timing analysis disfavour equally well-fit double temperature blackbody models where both the star surface and small hotspots contribute to the X-ray flux of the source. We identified a low significance ($1sigma$) temporal trend on the parameters of the source in the analysis of RGS data dating back to 2002, which may be explained by unaccounted calibration issues and spectral model uncertainties. The new dataset also shows no evidence of the previously reported narrow absorption feature at $sim570$ eV, whose possible transient nature disfavours an atmospheric origin.
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