Para fazer funcionar o método POST deve ser feito idem o GET deve fazer o parse da URI e depois passar o baseUrl e em seguida o headers e o Body.

Future<Transaction> save(Transaction transaction) async {
  final Map<String, dynamic> transactionMap = {
    'value' : transaction.value,
    'contact' : {
      'name' : transaction.contact.name,
      'accountNumber' : transaction.contact.accountNumber,
    }

  };
  final String transactionJson = jsonEncode(transactionMap);

  final Response response = await client.post(Uri.parse(baseUrl), headers: {
    'Content-type': 'application/json',
    'password': '1000',
  }, body: transactionJson
  );
  Map<String, dynamic> json = jsonDecode(response.body);
  final Map<String, dynamic> contactJson = json['contact'];
  return Transaction(
    json['value'],
    Contact(
      0,
      contactJson['name'],
      contactJson['accountNumber'],
    ),
  );
  return transaction;
}

Fazendo isso o código já ira compilar.

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