I have received the new printer and it does not shut down by itself so the other one is confirmed to be a malfunction. After receiving the printer, I used the printer menu to set the IP address and left all the software the same so no settings were changed.
However, this printer was not without it's own issue. When attempting to access the printer using a browser, it displayed a page titled "Home Update Page" where it was requesting an upload file. After some thought I decided to reflash the printer's network card firmware even though it reported it had the current version (1.03). After the reflash, browser access was normal so there must have been an incorrect or corrupt copy of the html files on the printer.
Overall I'm pleased with Epson's handling of the issue but be sure to check your printer thorougly.