After doing some tests where I put the Utilities.Serialize... code into the working FullName, that worked.  Then I renamed FullName to FullNameXX and it failed.  After a few more tests I came to the conclusion that all of the properties to be saved in the Profile need to be strings (or Binary I assume but I am not using the PropertyValuesBinary data field in SQL.)  So when I had a complex field, like my List then I had a property for the programmers that gets and sets a List as well as a string version of the same property that gets stored.  So I now have a SectionSettings and a SectionSettingsValue property.  Another adjustment needed was to make it only a DataContract and NOT Serializable or you will get extra values like k__backingField which IMHO looks bad.
The following should be the complete code...
Here is my DataContract:
[DataContract]
public class UserProfileContract : ProfileBase
{
    #region Constructors
    public UserProfileContract()
    {
    } // UserProfileContract - Constructor
    public UserProfileContract(List<SettingSection> SectionSettings)
    {
        this.SectionSettings = SectionSettings;
    } // UserProfileContract - Constructor
    #endregion Constructors
    public static UserProfileContract CurrentUser
    {
        get { return (UserProfileContract)(ProfileBase.Create(Membership.GetUser().UserName)); }
    }
    public string FullNameValue { get; set; }
    public string SectionSettingsValue { get; set; }
    [DataMember(Name = "FullName")]
    public string FullName
    {
        get { return ((string)(base["FullNameValue"])); }
        set {
            base["FullNameValue"] = value;
            Save();
        }
    } // FullName - Property
    [DataMember(Name = "SectionSettings")]
    public List<SettingSection> SectionSettings
    {
        get { return Utilities.Deserialize<List<SettingSection>>(base["SectionSettingsValue"].ToString()); }
        set
        {
            base["SectionSettingsValue"] = Utilities.Serialize<List<SettingSection>>(value);
            Save();
        }
    } // SectionSettings - Property
} // UserProfileContract - Class
[DataContract]
public class SettingSection
{
    public SettingSection()
    {
        this.UserSettings = new List<UserSettingPair>();
    } // SettingSection - Constructor
    public SettingSection(List<UserSettingPair> UserSettings)
    {
        this.UserSettings = UserSettings;
    } // SettingSection - Constructor
    [DataMember]
    public string SectionName { get; set; }
    [DataMember]
    public List<UserSettingPair> UserSettings { get; set; }
} // SettingSection - Class
[DataContract]
public class UserSettingPair
{
    [DataMember]
    public string Key { get; set; }
    [DataMember]
    public string Value { get; set; }
} // UserSettingPair - Class
Here is my static Utilities class:
public static T Deserialize<T>(string json)
{
    var obj = Activator.CreateInstance<T>();
    if (string.IsNullOrWhiteSpace(json))
        return obj;
    using (var ms = new MemoryStream(Encoding.Unicode.GetBytes(json)))
    {
        var serializer = new DataContractJsonSerializer(obj.GetType());
        obj = (T)serializer.ReadObject(ms);
        return obj;
    } // using the memory stream
} // Deserialize - Method
public static string Serialize<T>(object input)
{
    string Result = "";
    DataContractJsonSerializer ser = new DataContractJsonSerializer(typeof(T));
    using (MemoryStream ms = new MemoryStream())
    {
        ser.WriteObject(ms, input);
        Result = Encoding.Default.GetString(ms.ToArray());
    }
    return Result;
} // Serialize - Method
Here is an example of how to Save data in multiple sections as well as the FullName property:
// First Section
SettingSection s = new SettingSection();
s.SectionName = "ContractSearch";
UserSettingPair usp = new UserSettingPair();
usp.Key = "Default Control";
usp.Value = "txtFirstTextBox";
s.UserSettings.Add(usp);
usp = new UserSettingPair();
usp.Key = "Field1Choice";
usp.Value = "SchoolName";
s.UserSettings.Add(usp);
List<SettingSection> ss = new List<SettingSection>();
ss.Add(s);
// Second Section
s = new SettingSection();
s.SectionName = "Workflow Settings";
usp = new UserSettingPair();
usp.Key = "Primart Thing";
usp.Value = "Blabla bla";
s.UserSettings.Add(usp);
usp = new UserSettingPair();
usp.Key = "Allowable Tries";
usp.Value = "3";
s.UserSettings.Add(usp);
usp = new UserSettingPair();
usp.Key = "Extra Value";
usp.Value = "Gigity";
s.UserSettings.Add(usp);
ss.Add(s);
UserProfileContract.CurrentUser.FullName = "Grigsby";
UserProfileContract.CurrentUser.SectionSettings = ss;
Here is an Extension Method I created to make extraction of SectionSetting values easier:
public static T GetSectionValue<T>(this UserProfileContract up, string Section, string Property)
{
    string value = (from ss in up.SectionSettings
                                        from us in ss.UserSettings
                                        where ss.SectionName == Section
                                        && us.Key == Property
                                        select us.Value).FirstOrDefault();
    try
    {
        return (T)Convert.ChangeType(value, typeof(T));
    }
    catch (InvalidCastException)
    {
        return default(T);
    }
} // GetSectionValue - Extension Method
And lastly, an example of the above Extension method:
string k = x.GetSectionValue<string>("Workflow Settings", "Primary Thing");
string g = x.GetSectionValue<string>("Workflow Settings", "Extra Value");
int three = x.GetSectionValue<int>("Workflow Settings", "Allowable Tries");
Here is the string version of the values I put in:
[{"SectionName":"ContractSearch","UserSettings":[{"Key":"Default Control","Value":"txtFirstTextBox"},{"Key":"Field1Choice","Value":"SchoolName"}]},{"SectionName":"Workflow Settings","UserSettings":[{"Key":"Primart Thing","Value":"Blabla bla"},{"Key":"Allowable Tries","Value":"3"},{"Key":"Extra Value","Value":"Gigity"}]}]Grigsby
In the above string k = ... example, note that it returns null because the data is "Primart Thing" and not "Primary Thing".
Hope this helps someone out there.