R – The name xxx does not exist in the current context. What am I doing wrong

Good reason for that - you can't share objects across different DataContexts.

If you use the LinqDataSource's OnContextCreating event you can attach the correct instance to the data source [ie, that which the second entity is retrieved from] so they're both from the same DataContext:

ASPX:

<asp:LinqDataSource .... OnContextCreating="GetDataContext" ... />

C#:

protected void GetDataContext(object sender, LinqDataSourceContextEventArgs e)
{
    e.ObjectInstance = MyDataContext.Current; // assuming that Current is the ambient instance
}

The trick for editing is that you don't want to update every number, only the numbers affected by the change. For one item it can be easy, but if you are editing multiple items at once I would store the items in a List sorted by SortOrder:

Dim products = (From p As Product In _db.Products _
        Order By p.SortOrder _
        Select p).ToList()

Then I would rearrange the list according to the new values:

// do a RemoveAt first if you need to move an item
products.InsertAt(sortOrder, dic); 

Then renumber everything (or everything starting with SortOrder if you want to optimize):

Dim index As Integer = 0
For Each p As Product In products
    p.SortOrder = index
    index += 1
Next

If you do all your rearranging in one block then do the update in one block you'll save some redundant work.

This approach won't work so well if you have a large recordset. In that case you'll need to work out a more efficient approach, such as by using the database to do your work for you.

DECLARE @sortOrder INT
DECLARE @maxSortOrder INT
SET @sortOrder = 5
SET @maxSortOrder = 10
UPDATE Products SET SortOrder = SortOrder + 1 
WHERE SortOrder >= @sortOrder AND SortOrder < @maxSortOrder