The fully heavy axial-vector diquark-antidiquark structures $bb\bar{c}\bar{c}$ are explored by means of the QCD sum rule method. They are modeled as four-quark mesons $T_1$ and $T_2$ composed of $b^{T}C{\sigma }_{\mu \nu }{\gamma }_{5}b$, $\bar{c}{\gamma }^{\nu }C{\bar{c}}^T$ and $b^{T}C{\gamma }_{\mu }{\gamma }_{5}b$, $\bar{c}C{\bar{c}}^T$ diquarks, respectively. The spectroscopic parameters of the tetraquarks $T_1$ and $T_2$ are determined in the context of the QCD two-point sum rule method. Results obtained for masses of these states $m_1=(12715\pm 86)\mathrm{MeV}$ and $m_2=(13383\pm 92)\mathrm{MeV}$ are used to fix their strong decay channels. The full width $\Gamma \left(T_1\right)$ of the diquark-antidiquark state $T_1$ is estimated by considering the processes $T_1\to B_c^{-}{B}_c^{⁎-}$ and $T_1\to B_c^{⁎-}{B}_c^{⁎-}$. The decays to mesons $B_c^{-}{B}_c^{⁎-}$, $B_c^{-}\left(2S\right)B_c^{⁎-}$ and $B_c^{⁎-}{B}_c^{⁎-}$ are employed to evaluate $\Gamma \left(T_2\right)$. Results obtained for the widths $\Gamma \left(T_1\right)=(44.3\pm 8.8)\mathrm{MeV}$ and $\Gamma \left(T_2\right)=(82.5\pm 13.7)\mathrm{MeV}$ of these tetraquarks in conjunction with their masses are useful for future experimental studies of fully heavy resonances.