There are nuances between them:

Tergiverser: hesitate, waver on a decision

Atermoyer: Same, but formal, literary

Louvoyer: Do not move straight, weave, not necessarily figuratively.

Tataouiner: French Canadian specific.

Zigonner: Idem

Biaiser: To skew, go a circuitous way.

Tergiverser is by far the most common/neutral term for this, if a little formal. It’s not quite neutral, being somewhat derogatory I believe.

Atermoyer is a lot more literary (and as such not derogatory), but seems to have acquired a pronominal construction s’atermoyer sur probably under the influence of s’attarder. This pronominal construction feels a lot less literary to me. I’ve sometimes seen it used with the meaning “wallow over (smth)”. Otherwise, atermoyer mostly a synonym of tergiverser, but also appears used in the meaning of “deviate on a different topic”.

Zigonner (intr.) has the literal meaning of “fiddle with something” usually with the implication of pointlessness, either because the actions itself are pointless (think fidget toys) or the goal specifically won’t be reached that way (if ever at all). Hence by extension “waste time” (here through hesitation), but the literal meaning is more common overall.

Tataouiner is more specifically close to tergiverser,but more openly derogatory about it. Generally mildly insulting.

Louvoyer implies intent. Tergiverser is general hesitation (whether intended or not to dodge), but louvoyer (to me) says you are very much willfully dodging having to make a decision or answer a question.

Biaiser doesn’t actually have a meaning of hesitation as far as I know. I’ve personally never heard it used in that way. It’s sometimes used to translate the english to bias, especially in adjectival form, though.

Tataouiner and zigonner are quebecois stuff: I’ve never heard any of them in France.

Atermoyer is not really used anymore but the noun atermoiement is:

“Cessez vos atermoiements et continuons” but it is definitely very formal language.

Tergiverser and biaiser are common but I’d say that tergiverser is used by educated people while biaiser can be heard anywhere:

“Il a encore essayé de biaiser mais il m’aura pas cette fois.”

louvoyer is a bit like biaiser: it is nautical vocabulary so I’d say it is fairly popular and common.