What's the best way to convert dtsx packages to query a new server?
Two options come to mind:

If your dtsx packages are stored in the file system, you could use POWERSHELL  to read in each file, search and replace the connection string and rewrite the file.  Here is an example: 

(Make sure you make a copy of the folder BEFORE running the Powershell script, just in case):

$DtsxFiles = Get-ChildItem C:\Users\uswsh\Documents\*.dtsx -rec
foreach ($file in $DtsxFiles)
{
    (Get-Content $file.PSPath) |
    Foreach-Object { $_ -replace "localhost", "localhost1" } |
    Set-Content $file.PSPath
}


If your packages are not stored in the file system, you could dynamically generate CREATE SYNONYM commands that would point to the new location.

CREATE SYNONYM [dbo].[OriginalTable] FOR [RemoteServer].[RemoteDatabase].[RemoteSchema].[RemoteTable]