stocken
to falter
To briefly pause or hesitate while speaking, often out of uncertainty, nervousness, or shock.
Als sie die schlechte Nachricht überbrachte, stockte ihre Stimme.
When she delivered the bad news, her voice faltered.

to falter / stall
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to falter
To briefly pause or hesitate while speaking, often out of uncertainty, nervousness, or shock.
Als sie die schlechte Nachricht überbrachte, stockte ihre Stimme.
When she delivered the bad news, her voice faltered.

'Stocken' (to falter/stall) is an intransitive verb that does not take an accusative object. The subject of the sentence is always the person, thing, or process that starts to falter or stall.
Mitten im Satz stockte der Redner.
In the middle of the sentence, the speaker faltered.
You can recognize the meaning from the subject: If a person, voice, or speech 'stockt' (falters), it means 'zögern' (to hesitate). If traffic, the economy, or negotiations 'stocken' (stall), it means 'zum Stillstand kommen' (to come to a standstill).
Ihre Stimme stockte, als sie von dem Ereignis erzählte.
Her voice faltered as she talked about the event.
In the meaning 'langsamer werden' (to slow down), 'stocken' (to stall) is often used with processes or movements. Typical subjects are 'der Verkehr' (traffic), 'die Verhandlungen' (negotiations), 'der Handel' (trade), or also 'der Atem' (breath).
Wegen des starken Schneefalls stockte der Verkehr auf allen Hauptstraßen.
Because of the heavy snowfall, traffic stalled on all main roads.
The idiom 'jemandem das Blut in den Adern stocken lassen' (to make someone's blood run cold) is very common. The person who is frightened is in the dative.
Der unheimliche Schrei ließ mir das Blut in den Adern stocken.
The eerie scream made my blood run cold.
to hang
'hängen' and 'stocken' can both mean that something is not continuing properly. 'hängen' is used colloquially especially for technology, for example when a computer or program suddenly stops responding. 'stocken' fits better with a process, movement, or sequence that slows down or briefly comes to a stop. With software, 'hängen' sounds more natural; with negotiations, traffic, or work steps, 'stocken' is more likely.

Nach dem Streit stockt die Arbeit im Team deutlich.
After the argument, the team’s work is noticeably stalling.
Mein Laptop hängt seit zehn Minuten, und ich kann nichts mehr anklicken.
My laptop has been frozen for ten minutes, and I can’t click anything anymore.