Studies on double beta decay processes in $^{106}$Cd were performed by using a cadmium tungstate scintillator enriched in $^{106}$Cd at 66% ($^{106}$CdWO$_4$) with two CdWO$_4$ scintillation counters (with natural Cd composition). No effect was observed in the data accumulated over 26033 h. New improved half-life limits were set on the different channels and modes of the $^{106}$Cd double beta decay at level of $lim T_{1/2}sim 10^{20}-10^{22}$ yr. The limit for the two neutrino electron capture with positron emission in $^{106}$Cd to the ground state of $^{106}$Pd, $T^{2 umathrm{EC}beta^+}_{1/2}geq2.1times 10^{21}$ yr, was set by the analysis of the $^{106}$CdWO$_4$ data in coincidence with the energy release 511 keV in both CdWO$_4$ counters. The sensitivity approaches the theoretical predictions for the decay half-life that are in the range $T_{1/2}sim10^{21}-10^{22}$ yr. The resonant neutrinoless double-electron capture to the 2718 keV excited state of $^{106}$Pd is restricted at the level of $T^{0 umathrm{2K}}_{1/2}geq2.9times10^{21}$ yr
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