The computer I installed TS3 on isn connected to the internet. How do I get downloaded items from the Exchange/EA store?
You can’t. The Exchange and EA store run the Launcher directly to install their files. You can’t download the raw files at all, which you’d have to be able to do to move them to another computer. The only option would be to install TS3 on the computer you can connect on, download the sims3pack files, and transfer them over.