A label can be on a thread due to being on a single message in said thread. Your code simply goes label -> all label threads -> all thread messages, rather than accessing only the messages in a thread with a given label. That's not really your fault - it's a limitation of the Gmail Service. There are two approaches that you can use to remedy this behavior:
The (enable-before-use "advanced service") Gmail REST API
The REST API supports detailed querying of messages, including per-message label status, with Gmail.Users.Messages.list and the labelIds optional argument. For example:
// Get all messages (not threads) with this label:
function getMessageIdsWithLabel_(labelClass) {
  const labelId = labelClass.getId();
  const options = {
    labelIds: [ labelId ],
    // Only retrieve the id metadata from each message.
    fields: "nextPageToken,messages/id"
  };
  const messages = [];
  // Could be multiple pages of results.
  do {
    var search = Gmail.Users.Messages.list("me", options);
    if (search.messages && search.messages.length)
      Array.prototype.push.apply(messages, search.messages); 
    options.pageToken = search.nextPageToken;
  } while (options.pageToken);
  // Return an array of the messages' ids.
  return messages.map(function (m) { return m.id; });
}
Once using the REST API, there are other methods you might utilize, such as batch message label adjustment:
function removeLabelFromMessages_(messageIds, labelClass) {
  const labelId = labelClass.getId();
  const resource = {
    ids: messageIds,
    // addLabelIds: [ ... ],
    removeLabelIds: [ labelId ]
  };
  // https://developers.google.com/gmail/api/v1/reference/users/messages/batchModify
  Gmail.Users.Messages.batchModify(resource, "me");
}
Result:
function foo() {
  const myLabel = /* get the Label somehow */;
  const ids = getMessageIdsWithLabel_(myLabel);
  ids.forEach(function (messageId) {
    var msg = GmailApp.getMessageById(messageId);
    /* do stuff with the message */
  });
  removeLabelFromMessages_(ids, myLabel);
}
Recommended Reading:
Tracked Processing
You could also store each message ID somewhere, and use the stored IDs to check if you've already processed a given message. The message Ids are unique.
This example uses a native JavaScript object for extremely fast lookups (vs. simply storing the ids in an array and needing to use Array#indexOf). To maintain the processed ids between script execution, it uses a sheet on either the active workbook, or a workbook of your choosing:
var MSG_HIST_NAME = "___processedMessages";
function getProcessedMessages(wb) {
  // Read from a sheet on the given spreadsheet.
  if (!wb) wb = SpreadsheetApp.getActive();
  const sheet = wb.getSheetByName(MSG_HIST_NAME)
  if (!sheet) {
    try { wb.insertSheet(MSG_HIST_NAME).hideSheet(); }
    catch (e) { }
    // By definition, no processed messages.
    return {};
  }
  const vals = sheet.getSheetValues(1, 1, sheet.getLastRow(), 1);
  return vals.reduce(function (acc, row) {
    // acc is our "accumulator", and row is an array with a single message id.
    acc[ row[0] ] = true;
    return acc;
  }, {});
}
function setProcessedMessages(msgObject, wb) {
  if (!wb) wb = SpreadsheetApp.getActive();
  if (!msgObject) return;
  var sheet = wb.getSheetByName(MSG_HIST_NAME);
  if (!sheet) {
    sheet = wb.insertSheet(MSG_HIST_NAME);
    if (!sheet)
      throw new Error("Unable to make sheet for storing data");
    try { sheet.hideSheet(); }
    catch (e) { }
  }
  
  // Convert the object into a serializable 2D array. Assumes we only care
  // about the keys of the object, and not the values.
  const data = Object.keys(msgObject).map(function (msgId) { return [msgId]; });
  if (data.length) {
    sheet.getDataRange().clearContent();
    SpreadsheetApp.flush();
    sheet.getRange(1, 1, data.length, data[0].length).setValues(data);
  }
}
Usage would be something like:
function foo() {
  const myLabel = /* get label somehow */;
  const processed = getProcessedMessages();
  myLabel.getThreads().forEach(function (thread) {
    thread.getMessages().forEach(function (msg) {
      var msgId = msg.getId();
      if (processed[msgId])
        return; // nothing to do for this message.
      processed[msgId] = true;
      // do stuff with this message
    });
    // do more stuff with the thread
  });
  setProcessedMessages(processed);
  // do other stuff
}
Recommended Reading: