The major mistake is considering a nucleus like if it were an elementary particle.

Compare a nucleus spin configuration to spin electronic configuration of atoms. In both cases, there are multiple fermions (nucleons or electrons) that can have various total spín configuration. 

There are multiple known isotopes with known excited high spin nuclear isomers.

My favourite [isotopes of tantalum][1] show some of them have up to 7 isomers like $\ce{^{179}Ta}$, with spin 7/2, 9/2, 1/2, 21/2, 23/2, 25/2 and 37/2. 

One of knowm metastable nuclear isomers is even naturally occuring and  observationally stable: $\ce{^{180\mathrm{m}}Ta}$, even if the lower energy $\ce{^{180}Ta}$ is unstable and has the half-life about 8 hours, beta decaying to $\ce{^{180}Hf}$ or $\ce{^{180}W}$.

Military circles have been interested for some time in the metastable isomer [hafnium-178m2][2] with the half-life 31 years. The supposed stimulated gamma emission would provide about 1/100 energy density, compared to nuclear fission, while bypassing nuclear weapon regulations, as seen in [hafnium controversy][3].


  [1]: https://en.wikipedia.org/wiki/Isotopes_of_tantalum
  [2]: https://en.wikipedia.org/wiki/Isotopes_of_hafnium
  [3]: https://en.wikipedia.org/wiki/Hafnium_controversy