schwanken
to sway
To move back and forth uncertainly or irregularly; to wobble.
Die alte Hängebrücke schwankte bedrohlich im Wind.
The old suspension bridge swayed ominously in the wind.

to sway / fluctuate / waver
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to sway
To move back and forth uncertainly or irregularly; to wobble.
Die alte Hängebrücke schwankte bedrohlich im Wind.
The old suspension bridge swayed ominously in the wind.

The core idea of 'schwanken' (to sway/fluctuate/waver) is always some form of instability or irregularity. This idea connects all meanings: a physically swaying bridge, unstable prices, or a person who cannot make a decision.
Das Boot schwankte stark auf den Wellen.
The boat swayed strongly on the waves.
For physical movement, objects such as 'Brücke' (bridge), 'Baum' (tree), or 'Schiff' (ship) are often used as the subject. But it can also be a person, for example someone who is drunk or dizzy.
Der Betrunkene schwankte die Straße entlang.
The drunk man staggered down the street.
When talking about values or amounts that change, the structure 'schwanken zwischen ... und ...' (to fluctuate between ... and ...) is often used to indicate the range of the change.
Die Benzinpreise schwanken täglich zwischen 1,80 € und 1,90 €.
Gas prices fluctuate daily between €1.80 and €1.90.
To express a person's indecision, the patterns 'schwanken zwischen [Option A] und [Option B]' (to waver between [Option A] and [Option B]) or the subordinate-clause construction 'schwanken, ob ...' (to waver over whether ...) are often used.
Sie schwankt noch, ob sie den neuen Job annehmen soll.
She is still wavering over whether she should accept the new job.
In the meaning 'not to be constant', 'schwanken' (to vary) is frequently combined with the nouns 'Leistungen' (performances) or 'Qualität' (quality). The degree of change is often described with an adverb such as 'stark' (greatly/strongly).
Die Qualität der Produkte schwankt leider stark.
Unfortunately, the quality of the products varies greatly.
to change / switch
'wechseln' and 'schwanken' can both describe that something does not stay the same. 'wechseln' usually means a clear transition from one thing to another, for example from one topic or state to another. 'schwanken' more often means that something goes back and forth uncertainly or irregularly, such as prices, opinions, or performance. You can confuse the words when something changes several times. In that case, 'wechseln' sounds more neutral and orderly; 'schwanken' emphasizes that something is not stable.

Die Preise für Benzin schwanken diese Woche stark.
Gasoline prices are fluctuating sharply this week.
Im Gespräch wechselt Mehmet plötzlich das Thema.
During the conversation, Mehmet suddenly changes the topic.