The major mistake is considering 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 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.
Some of them are even observationally stable like $\ce{^{180m}Ta}$, even if $\ce{^{180}Ta}$ has the half-life about 8 hours.