Well when using origin as long as you sign into the same account as you did on the other computer it will work but the data on the other computer swill not be the same.
I don't know how you got sims 4 without signing into origin.... On your first pc..... Your origin is the same thing as your ea login though.....so thats your problem not mine
So what you have to do is sign into origin/make an account, then you will be able to go offline. you have to sign in and make sure the game is working before you can go offline. you will be able to go offline if you click origin on the top left side and click offline mode. there is no way around not signing into origin right now but im sure as time goes on people will find ways to get around to it. Almost all new EA games are linked to origin.
You've probably sorted this by now, so may ignore my response if you have.
When buying anything online, card details are not the only way to pay. When the payment screen comes up on Origin, there should be multiple tabs at the top with different options for how to pay. The one that I always use is the EA Wallet function.
This is where I go into my local Game shop to buy a card or two for amounts that add up to the cost of what I want to buy. These cards work similarly to vouchers or other such cards like Nintendo eShop credit. You can pay by cash this way, but it does not mean you are buying the game itself; you are buying the equivalent credit.
After you have bought this card, you can take it home with you to redeem the money you just paid. There will be a section on the back which you scratch off to reveal the code. There are also instructions to follow about how you redeem this credit using the code.
Once the credit has been added to your account, you can buy the games/dlc you want.
Alternatively, your Game shop might also sell a code card for the specific content or game; which is definitely a lot simpler.
The Sims 4 Base Game also comes in disc form to install, but I believe you need Origin to be running in the background. Don't quote me on this specific part, though, as I installed my game over a year ago. I have been hooked since.
Also, here is what the Origin Cards look like for reference. My finger is covering the location of the code. I thought I'd show the front and back of the same card. There may be ones you can get for £10 still as I used to buy two at any one time to buy two stuff packs together. This is the latest one I got, because it makes more sense and I was able to take advantage of the recent sale combined with remaining funds I already had on my account.