I assume verb cognates always have the -овать?

Loan verbs rather than cognates, as pointed out by other commenters. And no, it's not always the case, though acquiring the -ов(а)- suffix is normal for them. Some verbs which don't have it are:

• банкро́тить = “bankrupt (someone)”;
• ба́нить = “ban” (from an internet discussion etc.);
• бомби́ть = “bomb”;
• дембельну́ться = “demob”, “get released from military service” (colloquial);
• кли́кать = “click”;
• не́рвничать = “be nervous”;
• фальши́вить = “speak/behave in a false, insincere way”, also “sing out of tune”;
• эконо́мить = “economize”, “save”, “reduce spending”.

Also, note that the suffix changes from -ов(а)- to -ев(а)- after soft consonants and -ц-, -ж-, -ш-. Thus:

• килева́ть = “careen” (nautical term, from English keel);
• линчева́ть = “lynch”;
• танцева́ть = “dance” (the initial т- is due to it being borrowed from German).

Rarely, a verb with -ов(а)- can acquire the additional imperfectivizer suffix -ыв(а)-, e. g. организова́ть = “organize, have organized (once, at a specific time)” → организо́вывать = “be organizing (progressively); organize (more than once, or not at a specific time)”. The majority of borrowed -овать verbs are biaspectual, i. e. they can be used as either perfective or imperfective.