wanken
to wobble
To move unsteadily back and forth because one is losing support or balance. Describes physical instability.
Nach dem Karussell wankte er ein wenig, als er ausstieg.
After the carousel, he wobbled a little as he got off.

to wobble / waver
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to wobble
To move unsteadily back and forth because one is losing support or balance. Describes physical instability.
Nach dem Karussell wankte er ein wenig, als er ausstieg.
After the carousel, he wobbled a little as he got off.

'Wanken' (to wobble/waver) usually forms the perfect tense with 'haben' when the movement takes place in one location. The perfect tense with 'sein' is used when movement from one place to another is being described.
Der alte Schrank hat bei der Berührung gewankt.
The old cabinet wobbled when it was touched.
The noun corresponding to this verb is 'das Wanken' (wobbling/wavering). It is often used in the fixed expressions 'ins Wanken geraten' (to become unstable) or 'etwas ins Wanken bringen' (to make something unstable).
Durch die Krise geriet das ganze Finanzsystem ins Wanken.
The crisis made the whole financial system start to totter.
In the figurative meaning, 'wanken' (to waver) refers to abstract things such as convictions, decisions, principles, or loyalty. You can recognize this meaning from the context of doubt or uncertainty.
Sie ließ sich nicht beirren und wankte nicht in ihrer Überzeugung.
She did not let herself be unsettled and did not waver in her conviction.
When 'wanken' (to totter) is used for structures, it often refers to large, powerful systems. Typical subjects are 'die Regierung' (the government), 'der Thron' (the throne), 'ein Imperium' (an empire), or 'ein Unternehmen' (a company).
Nach dem Rücktritt des Präsidenten wankte die gesamte Regierung.
After the president's resignation, the entire government tottered.
to doubt
'zweifeln' and 'wanken' can both express uncertainty. 'zweifeln' is used when someone is not sure whether something is true, right, or good. People often say 'an etwas zweifeln' (to doubt something). 'wanken' describes more that a firm opinion, decision, or loyalty becomes uncertain. It sounds more figurative: a person or position is no longer steady. The words can be confused because both show inner uncertainty. 'zweifeln' is more natural when you are talking about doubts or concerns. 'wanken' fits better when a previously firm conviction suddenly becomes uncertain.

Nach den neuen Informationen wankt Tara in ihrer Meinung.
After the new information, Tara wavers in her opinion.
Noah zweifelt nach dem Gespräch an seiner Entscheidung.
After the conversation, Noah doubts his decision.