Thank you, Katie, for the suggestion! I think missing data was the issue with the Stata format.
We found a way to successfully import data into SPSS. Export the data file from EpiData Manager as a .csv file (I used the Linux setting). Using Excel, I opened a blank spreadsheet and imported the .csv file via the "add from text" option under "data". I specified that it is delimited by commas, and then changed each column from "general" format to "text" format. This was tedious but necessary because Excel automatically deletes any leading 0s upon import. If you are using these notes as instructions, DO NOT CLICK IN ANY CELL. It will revert to general format, and delete leading 0s. I saved the file as a .txt file (using MS-DOS format). Then I opened in Microsoft Word. I deleted the variable names. I then used find and replace to find "^t" and replace with "^p". SPSS doesn't like tabs, so I needed to replace all tabs with paragraphs.
You can use the syntax EpiData Manager produces when exporting to SPSS, I think. Or write your own. I wrote my own, because, as noted earlier in this thread, I had variables embedded within the lines.
Worked like a charm!