The best replacement for affidavit is declaration.

The best alternatives to affidavit are rated as following:
★★★★★  declaration
★★★  affirmation
★★  deposition
★  sworn statement
★  statutory declaration
★  statement
★  certification
★  affidavit
★  declaration federal court
★  notary
  unsworn declaration
  acknowledgement
  notarized statement
  stat dec
  attestation
  witness statement
  verification
  acknowledgment
  power of attorney
  oath
  bill of sale
  sworn declaration

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Speaking about affidavit, one probably shouldn't forget about jurat.

Other search terms related to affidavit and their ratings are:
★★★★★ jurat
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affidavit
declaration
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