Can't figure out last part of a powershell script

Question

Hey guys,
First time posting here. I have a powershell project for school that I'm trying to figure out. I don't have a lot of experience with powershell, but I've gotten most of my project done. I am having issues with this last part. Here is the part of the assignment
I can't seem to get working properly:
Process the items in the <localgroupManagement> node.
Find all non-domain controller computers in the domain
Add the domain global group specified in the <members><group> node(s) to each discovered computer’s local group specified by the
<name> node.  If the local group does not exist create it.
Here is the code I currently have:
param([string]$filename)
Function createOU($OU) {
    if ( !(Get-ADOrganizationalUnit -filter {name -eq $OU}) ) {
        Write-Host "Creating new OU: $OU"
        New-ADOrganizationalUnit -Name $OU -Path $ADDomainName
    else {
        Write-Host "OU already exists: $OU"
Function addUser($account, $fn, $ln, $desc, $pwd, $OU, $ADDomainName, $ADgroup) {
    if ( !(Get-ADUser -Filter {name -eq $account}) ) {        
        Write-Host "Adding user: $account"
        New-ADUser $account -GivenName $fn -Surname $ln -Description $desc -Enabled 1 `
        -DisplayName ($fn + " " + $ln)  -AccountPassword $pwd -ChangePasswordAtLogon $true `
        -Path "OU=$OU,$ADDomainName"
    else {
        Write-Host "User already exists: $account"
Function addGroup($ADgroup, $ADDomainName, $OU) {
    if ( !(Get-ADGroup -Filter{name -eq $ADgroup}) ) {          
        Write-Host "Adding group: $ADgroup"
        New-ADGroup -Name $ADgroup -GroupCategory Security `
        -GroupScope Global -Path "OU =$OU,$ADDomainName" 
    else {
        Write-Host "Group already exists: $ADgroup"
### MAIN PROGRAM
if( !$filename ) {
    # prompt the user to select a file to process
    [System.Reflection.Assembly]::LoadWithPartialName("system.windows.forms")
    $dialog = New-Object System.Windows.Forms.OpenFileDialog
    $dialog.Filter = "XML Files (*.xml)|*.xml|All Files(*.*)|*.*"
    $dialog.Title = "Select an XML file to import“
    $result = $dialog.showdialog()
    if ( $result -eq "ok" ){
        $filename = $dialog.Filename
$userfile = [xml](Get-Content $filename)
$ADDomainName = (Get-ADDomain).DistinguishedName
foreach( $user in $userfile.root.userManagement.user ) {
    $account = $user.account
    $fn = $user.firstname
    $ln = $user.lastname
    $desc = $user.description
    $pwd = ConvertTo-SecureString -AsPlainText $user.password -force
    $mgr = $user.manager
    $OU = $user.ou
    Write-Host
    Write-Host $account
    Write-Host $fn
    Write-Host $ln
    Write-Host $desc
    Write-Host $user.password
    Write-Host $mgr
    Write-Host $OU
    createOU $OU $ADDomainName
    foreach( $groupmem in $user.memberof.group ) {
        $ADgroup = $groupmem
        addGroup $ADgroup $ADDomainName $OU
        addUser $account $fn $ln $desc $pwd $OU $ADDomainName $ADgroup
        Add-ADGroupMember $ADgroup -Members $account      
#Local Management
$computers = Get-ADComputer -Filter * | where {-not($_.distinguishedname -like "*OU=Domain Controllers*")}| `
foreach{$_.name}
Write-Host "Computers that already exsist on the Network:"
foreach($computer in $computers) {
$DomainName =(Get-ADDomain).NetBIOSName
$localgroups = $userfile.root.localGroupManagement.localGroup
    foreach($lg in $localgroups.name) {
        try {   
            $newgrp = [adsi]::Exists("WinNT://$computer/$lg,group")
            if($newgrp -eq $true) {
                Write-Host "$lg Already Exists!"
            else {
           $ds=[adsi]("WinNT://$computer,computer")
           $group=$ds.create("group",$lg)
                $group.setinfo() 
        catch [Exception] {
write-host "Exception"
I have tried other methods, but this is the one I receive the least amount of errors on. The script starts to have issues here:   $newgrp = [adsi]::Exists("WinNT://$computer/$lg,group")
I have looked information up about WinNT Exists and it appears it has some issues. I have been trying to figure out how to add the domain groups to the local computer's groups. The tricky part is that the script has to run without knowing the computer's
name so it can run on any domain. If anyone could help me out with this last part I would be very grateful.

Answer 1

The script should be using WinNT to find computers on the domain and add domain group to each of those computers local group. I have provided portions of the xml file and instructions to help clarify.
Here is the portion of the xml file I'm having issues with
<localGroupManagement>
<localGroup>
<name>Power Users</name>
<members>
<group>Marx Brothers</group>
</members>
</localGroup>
<localGroup>
<name>Musicians</name>
<members>
<group>GGMusicians</group>
</members>
</localGroup>
</localGroupManagement>
</root>
Here is what the assignment is assuming.
Assumptions
The computer the script is running on will be in the domain that you wish to manage. 
The computer running the script will have the Active Directory PowerShell module installed.
The user running the script will have rights to do all required tasks