Competitive protonation between alkene and carbonyl
In the scheme below I have shown the three most probable protonations of the molecule in question. I have considered that a tertiary carbocation is typically more stable than a secondary one and that a resonance-stabilised carbocation is typically more stable than a non-stabilised one.



From left to right, let’s discuss the structures.


The one on the left is not good. The carbocation is only stabilised by being tertiary. It would take a very strong acid to sufficiently protonate in this way, or a good nucleophile that can capture the positive charge on the tertiary carbocation.
The one in the middle doesn’t look as bad initially, because it is resonance-stabilised. However, if you draw the other principal mesomeric structure, you arrive at a positively charged oxygen with a sextet — a very unstable thing to have. It is very improbable that protonation would occur like this.
There is another (undrawn) structure that would protonate the same double bond but on the other side. However, that carbocation would destroy the unsaturated ketone’s resonance and is therefore unfavored.
Finally, the protonated carbonyl. This is clearly the best cation, since it is an all-octet structure and the positive charge can be stabilised by resonance. Futhermore, if you draw the resonance structure, you realise that it localises the cation on a tertiary carbon — the best possible case.


Thus there should be no discussion; protonation of the carbonyl oxygen leads to the most stable cation and is therefore expected to prevail.

(Your lower $\mathrm{p}K_\mathrm{a}$ of carbocations is likely due to the fact that they can often deprotonate to form a double bond giving additional stabilisation.)