DropDownList variable cleared: I'm using the DropDownList command in one of my scripts to set dependent variable values.  The value selected in DropDownList variable 1 uses the Process command to set the selection list in DropDownList variable 2.  When I select the first dropdownlist the screen refreshes and clears the previously selected DropDownList variables.  How can I prevent this?

You have probably cleared the drop down list variables (long text variables) in your script yourself.
To avoid this effect, clear the variables only once for each transaction. For example:

if not V[_transactionid=&V[XD01_transactionid]]

    include "XD01_initialize.txt"

  endif

and in the include "XD01_initialize.txt":

// reset transaction variables

Set V[XD01_*] ""

// save transaction id

Set V[XD01_transactionid] "&V[_transactionid]"

// clear long text variables

Set text[XD01_DDL_SPRAS] ""

...

An additional tip. If you read the drop down list content via "Call" from the SAP system, you can use the new "cache" option of the Call command, e.g.

Call "Z_S10_SEARCHHELP" cache="session" in.SEARCHHELP="H_T002" in.COLUMNS="SPRAS(2),SPTXT(16)" table.DATA="data"

in order to save processing time (available in GuiXT 2011 Q4 1 and upwards).