VBA Range.FindNext just finds next cell, not next search term - excel

Question

I am still quite new to VBA and decided I would try and teach myself the Range.FindNext method. Unfortunately, I am not very successful so far.
What I am trying to do, is to copy all rows with a specific search term in them to a new sheet (could be anything, therefore declared as a Variant). What is important, is that the search term might only be part of the cell's value, hence I am using xlPart in my Range.Find method.
Here the example data from my ActiveWorkbook.ActiveSheet:
Date Name Numbers
12.04.2012 Marla 45653
13.04.2017 Peter 23545
27.04.1985 Bertrud 46932
16.08.2020 Peterson 46764
15.09.2014 Marcos 32465
21.06.2010 Peter Pan 23452
31.08.2013 Bernard 12321
So, when looking for "Peter", I should be getting rows 3, 5 and 7 in a new sheet. This is the code I wrote for this:
Option Explicit
Dim wsMain, wsNew As Worksheet
Dim rgAll, rgSearchTermFind As Range
Dim varSearchTerm As Variant
Dim lngLastRow, lngLastColumn As Long
Dim firstAddress As String
Public Sub FindAndCopy()
'I have an InputBox for the user to determine the varSearchTerm, but for this example:
varSearchTerm = "Peter"
Set wsMain = ActiveWorkbook.ActiveSheet
Set wsNew = Sheets.Add(After:=Worksheets(Sheets.Count))
Call FindLast(wsMain) 'This is a separate sub I wrote to find the last row & column
With wsMain
Set rgAll = .Range(.Cells(1, 1), .Cells(lngLastRow, lngLastColumn))
End With
With rgAll
Set rgSearchTermFind = .Find(What:=varSearchTerm, _
LookIn:=xlValues, _
LookAt:=xlPart, _
SearchOrder:=xlNext, _
MatchCase:=False)
If Not rgSearchTermFind Is Nothing Then
firstAddress = rgSearchTermFind.Address
Do
'Copy row to new sheet
If Application.WorksheetFunction.CountA(wsNew.Cells) <> 0 Then
Call FindLast(wsNew) 'This is a separate sub I wrote to find the last row & column
wsMain.Range(rgSearchTermFind.Address).EntireRow.Copy _
Destination:=wsNew.Cells(lngLastRow + 1, 1)
Else
wsMain.Range(rgSearchTermFind.Address).EntireRow.Copy _
Destination:=wsNew.Cells(1, 1)
End If
'Find next occurrence of search term
Set rgSearchTermFind = .FindNext(rgSearchTermFind)
Loop Until rgSearchTermFind.Address = firstAddress
Else
'Code here to execute if search term could not be found
End If
End With
End Sub
When running this code, the initial Range.Find method finds Peter in B3, but the Range.FindNext then finds "Bertrud" in B4 and copies it. This happens for each cell in the range, leaving me at the end with the table copied three times in the new sheet (due to there being three columns).
What am I doing wrong? Any help will be much appreciated.