Tetragonal double tungstate single crystals with formula Na$T$(WO${}_{4}$)${}_{2}$ have been grown by the Czochralski ($T$ = Gd, La, Y) or by the top-seeded solution growth ($T$ = Lu) methods with Tm concentration between 8 $\ifmmode\times\else\texttimes\fi{}$ 10${}^{18}$ and 7.85 $\ifmmode\times\else\texttimes\fi{}$ 10${}^{20}$ cm${}^{\ensuremath{-}3}$. The spectroscopic properties of Tm${}^{3+}$ in these crystals are related with the peculiarities of their $I\overline{4}$ crystalline structure. Sixty-five percent of La ions in NaLa(WO${}_{4}$)${}_{2}$ are in the 2$d$ site, while in the other crystal hosts, the lanthanide occupies preferentially the 2$b$ site (59$%$ in $T$ = Gd, 74$%$ in $T$ = Y, and 58$%$ in $T$ = Lu). As a consequence, the linewidths of spectral bands associated with the electronic transitions are significantly narrower in NaLa(WO${}_{4}$)${}_{2}$ than in the rest of the isostructural crystals considered. Polarized spectroscopic measurements at 5 K and at higher temperatures, along with energy level simulation of the 4${f}^{12}$ configuration using a single-electron Hamiltonian, including free-ion and crystal field interactions, allowed us to determine the irreducible representation and energy of Stark levels up to the ${}^{3}$P${}_{0}$ multiplet and thus to obtain realistic partition functions ($Z$) used for emission cross-section calculations. In particular, for the ${}^{3}$F${}_{4}$(u) \ensuremath{\rightarrow} ${}^{3}$H${}_{6}$(l) laser transition at \ensuremath{\lambda} \ensuremath{\approx} 2 \ensuremath{\mu}m, this provides: Z${}_{\mathrm{l}}$/Z${}_{\mathrm{u}}$ = 1.436 ($T$ = Gd), 1.464 ($T$ = La), 1.448 ($T$ = Y), and 1.471 ($T$ = Lu). Radiative lifetimes calculated by the Judd-Ofelt and F\"uchtbauer-Ladenburg methods are in agreement and decrease in the following order $T$ = Gd, La, Y, and Lu, however, nonradiative losses are stronger for $T$ = Gd and La crystals; therefore, experimental lifetimes of ${}^{1}$D${}_{2}$, ${}^{1}$G${}_{4}$, ${}^{3}$H${}_{4}$, and ${}^{3}$F${}_{4}$ Tm${}^{3+}$ multiplets do not change too much with crystal host. For 4.68 at.$%$ Tm:NaY(WO${}_{4}$)${}_{2}$ crystal continuous-wave laser operation is obtained with \ensuremath{\approx}42$%$ of slope efficiency and a record (for this crystal class) tuning capability of \ensuremath{\lambda} = 1847--2069 nm. The broad bandwidths, \ensuremath{\Delta}\ensuremath{\lambda}${}_{\mathrm{FWHM}}$ > 20 nm, of the free-running laser emission are promising for ultrafast (fs) mode-locked laser operation near \ensuremath{\lambda} \ensuremath{\approx} 2 \ensuremath{\mu}m.