I was able to solve this by removing  the highlighted code in bold from code-behind of my custom module.  Once, I did that, then I see the move icon. But why the highlighted code was causing this is not understood by me.

Also, not clear on how to get the edit icon show up next to settings icon?

CODE-BEHINDas below

Imports System
Imports System.Web
Imports System.Web.UI
Imports System.Web.UI.WebControls
Imports DotNetNuke
Imports DotNetNuke.Security
Namespace Sunil.ExamTracker
    Public Class viewExamTracker
        Inherits DotNetNuke.Entities.Modules.PortalModuleBase   'System.Web.UI.UserControl

        Protected Sub Page_Init(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles MyBase.Init
            Dim x As Boolean = Me.IsEditable

            ' declare module actions
            Dim action As DotNetNuke.Entities.Modules.Actions.ModuleAction = New   DotNetNuke.Entities.Modules.Actions.ModuleAction(GetNextActionID)
            action.Title = "Add New Exam"
            action.CommandName = ""
            action.Url = EditUrl()
            action.Secure = SecurityAccessLevel.Edit
            action.Visible = True
            MyBase.Actions.Add(action)
            'MyBase.Actions.Add(GetNextActionID, "Add New Exam", "", "", "", URL:=EditUrl(), secure:=SecurityAccessLevel.Edit, Visible:=True)
        End Sub
        Protected Sub Page_Load(ByVal sender As Object, ByVal e As System.EventArgs) Handles Me.Load
            'Put user code to initialize the page here

            Dim objExamTracker As New ExamTrackerController
            Dim list As ArrayList

            If Not Page.IsPostBack Then
                list = objExamTracker.GetExamTrackerList()
                Me.grdExamTracker.DataSource = list
                Me.grdExamTracker.DataBind()
            End If
        End Sub

    End Class