@enchantress6274 If none of your Store content works on the new computer, try reinstalling it in a clean Sims 3 folder, as in, move the existing one out of Documents > Electronic Arts and onto the desktop and open the launcher to spawn a new one. This time around, download and install only a few items at a time, and worlds separately. As you've seen, the launcher doesn't handle large amounts of data very well.
As it happens, I downloaded the pet store last week, and it works fine in-game. So the content isn't broken; it's just a matter of convincing the launcher to install it properly. If you want to be extra careful, check the size of the highest-numbered .ebc file in ...The Sims 3 > DCCache after installing content. These files are called dcdbX.ebc, with X counting up from zero. (The files that end in .dbc are custom content.)
The launcher will create a new .ebc file when the current one gets to about 205,000 KB, and over about 250 MB, the game has trouble reading the files. So if your current file is 198 MB, it's best to install smaller sets or individual items until you get a new .ebc file, then install large items one or two at a time until you get close to the limit on the next file.
This process can be tedious, but on the other hand, you'll hopefully only need to do it once. And you can keep a backup of the Sims 3 folder with only your Store content installed in case you need to restore it later for some reason.