@Azaileah EA doesn't allow codes to be shared or transferred among accounts, although of course one household can share access to games through a single account. It sounds like you own the base game but not the EPs? If you installed it on your new computer through Origin, you'd be on patch 1.69 with its mandatory Origin tie-in no matter what else you did, and you could only play with packs that were registered to your account. Even if your relative signed in and installed the packs for you, you would still need to open and sign into Origin every time you wanted to play. So no, there's really no way to add those packs to your existing TS3 install.
Some people have remained on (or reverted to) an all-disc install and avoided patch 1.69 and Origin entirely. But that requires disc copies of every pack you want, as well as a base game disc manufactured before September 2012, when downloading Origin became part of the TS3 install process. So even if you bought an external disc drive, you wouldn't be able to add the packs on your relative's discs to your digital install.
I wish I could help more, but EA is pretty strict with its rules about who owns the rights to run its content.