I haven't done many of these, so I don't have a lot of experience. I don't know the exact mechanism of these reactions, so I cannot offer advice from that perspective. I do think the rates of these reactions should be different (most reactions are). Since fluorine is generally reasonably resistant to dehydrofluorination reactions, I am guessing that running the reaction for a shorter time (stop before no pressure change occurs if Parr reduction or if plotting rates at atmospheric, look for a break in the curve). If you are cleaving a benzyl ether or ester, than doing so in the presence of an acid will increase the debenzylation rate and suppress the dehydrofluorination reaction.
If this still fails and you can change your protecting group, then change mechanism. You can use a more electron rich benzyl group so you can more easily use acid or an oxidative cleavage (cerium). Alternately, you could also try TMSI. It also might work. You can still try other mechanisms on the benzyl group, acid, nucleophiles, halogenation, etc. No real recommendations without knowing the problem however.