I'd immediately switch to any podcast app that automatically skips ads like the Youtube Ad Skipper Chrome extension.
Get sponsor block if you want to get rid of these pesky sponsorship messages in YouTube videos too
There was some discussion about integrating SponsorBlock into AntennaPod (a popular open source podcast app) a while back, but it didn't go anywhere. Ajay (developer of SB) was very open to the idea, so anyone interested in developing ad skipping technology for podcasts should probably get in touch with them. I think getting a proof of concept podcast app that skips ads should not be all that difficult.
I wonder if you could train some machine learning model using the data from SponsorBlock and achieve good results on podcasts as well. That way you wouldn't be dependent on a crowdsourced online database for your offline listening. Alternatively, even creating a transscript using something like [1] and scanning for words like "sponsor", "ad" or specific company names might already be a good enough heuristic.