Help Please!!

Like most games, TS3 has two parts to it -- the program install (this includes the base game and any EP/SPs, but that is all) and the user game folder in Documents (this includes any store or custom content, saved sims, mods, saved games, anything else the player has added on).

The user game folder is easy to transfer over to a new computer if required. Or one could transfer elements of it to a new game folder if they want to do it that way. That is the folder that should be backed up.

Program installs cannot be copied over that way. If what your computer tech (shop) is doing is imaging your current drive and copying the entire thing, installs and everything, over to your new computer that probably will not work. Usually the goal in these instances is to preserve your documents and data, not program installs.

You will likely have to reinstall the games somehow. What kind of an install was this and if disc, were you the original owner? If you were, then they should be registered to you via the TS3 site (or can be now) and then just show up in your Origin account if it has the same user name/credentials. If they originally belonged to and were registered to someone else, then no Origin cannot be used in that way as game ownership is not transferable. If your TS3 and Origin accounts are not matching up because you have two different user ids, then only EA customer support can fix that for you by way of a real time conversation with a support agent -- they offer web chat or telephone callback, depending on your location and time of day.

Thank you @igazor for responding to my question.
The games were installed from the discs and I am the original owner. But when I do try to register the games, it tells me that the serial codes have already been used.

So can you not do another disc install on the soon to be acquired new computer?

If not, then presumably you were the prior user of the codes or for some reason the wrong codes are being used. EA Customer Support will hopefully be able to help you sort that out, they are the only ones with access to customer account records.