You probably won't need this step-by-step, but I'll tell you what I do so it's easy to keep track of and fairly secure.
I have one gift card I keep as my main. Let's say I've named it Diz. Whenever I get new Disney gift cards, I first add Diz to my (empty) account and then add the new cards as CaptainAmerica said above. You have to give them a name so I just use a number or letter or something because I'll be deleting them when I'm done. After adding each new one, I transfer its balance to Diz and then delete it. When I'm done transferring all of the balances of my new gift cards to Diz, I then delete Diz so there are no cards left on my account.
The reason for that is that for a while, there were many instances of people getting into others people's gift card accounts and transferring the balances onto their own cards. To be safe, I never leave a card on my account that has any money on it. Deleting the card from your account doesn't affect the balance that's on the card, it just takes the card off the web so it can't be hacked.
So let's say you have 4 gift cards you want to combine. We'll call your cards Main and your other cards 2, 3, and 4 (I write the name of my main card on it in sharpie so I know it's my main card). Add Main to the disneygiftcard account you've created. Then, add the others one by one, transferring their balances onto Main. Be sure to delete the empty cards 2, 3, and 4. You'll then have all of the combined balances on Main and $0 on 2, 3, and 4. At that point, I write the total dollar amount in sharpie on Main, delete it from my online account, and throw away the other cards. JUST BE SURE YOU DON'T THROW AWAY MAIN! As I use it, I update the balance on the card itself with a sharpie, crossing out the old one. I even bring a sharpie with me on my trips. Every so often I switch out for a new Main card.