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We investigate theoretical and observational aspects of a warm inflation scenario driven by the $β$-exponential potential, which generalizes the well-known power law inflation. In such a scenario, the decay of the inflaton field into radiation happens during the inflationary phase. In our study, we consider a dissipation coefficient ($Γ$) with cubic dependence on the temperature ($T$) and investigate the consequences in the inflationary dynamics, focusing on the impact on the spectral index $n_s$, its running $n_{run}$ and tensor-to-scalar ratio $r$. We find it possible to realize inflation in agreement with current cosmic microwave background data in weak and strong dissipation regimes. We also investigate theoretical aspects of the model in light of the swampland conjectures, as warm inflation in the strong dissipation regime has been known as a way to satisfy the three conditions currently discussed in the literature. We find that when $Γ\propto T^3$, the $β$-exponential model can be accommodated into the conjectures.