Here's exactly what I just did, and it seems to have worked perfectly:
Backup everything from the SD card to your computer, then shut the phone off and remove the SD card (or turn the phone off first and use a USB reader to copy the files, obviously).
Put the SD card into a USB reader on your computer so you can repartition it (I used EASEUS Partion Master Home Edition). Follow the instructions here:
SD card partitioning - XDA-Developers. I used ext2 for the second partition. (From what I understand, it's easier on the SD card and won't hurt its life as much as ext3 or ext4.) I have a 32GB card, so I made 1GB for the ext2 partition. That seems like more than plenty, but you could always do more (or less) depending on your situation.
Copy everything back to your SD card from your computer. Windows only recognizes the first partition on an SD card, so you should see only your FAT32 partition in Windows Explorer like before.
Before putting your SD card back in your phone (probably doesn't matter if it's before, but I waited), boot directly to recovery and wipe the Dalvik cache to clear out the files in /cache/dalvik-cache, which we no longer want there because they're going on sd-ext now (which will happen automatically with the SD card properly partitioned). Then power the phone back off. Do not boot Android yet or it will recreate the Dalvik cache on internal memory (no big deal, but the point is we're clearing it off of internal memory so we gain that space back--you could always do this later, too).
Once your SD card is repartitioned and your data is copied back onto it, put the SD card back in the phone and boot normally. It will recreate the Dalvik cache files and will take a few minutes. When it's done, you should be all set.
After it booted, I checked with Solid Explorer and can confirm the Dalvik cache files are now in /sd-ext/dalvik-cache and not in /cache/dalvik-cache.
This seems to have fixed the "Package file is invalid." error, for the most part.
I did get it once more on a sizable app, but moving some more apps to the SD card (the normal way, in the apps settings) fixed it. It seems you still need a decent amount of internal app storage space not to get the error, but it's definitely much better now that the /cache partition isn't full.
I have been incredibly pleased with this ROM so far, and now even much more so.