I am out of space on my HDD for XBOX 360. How to...

I am out of space on my HDD for my Xbox 360, and I already purchased an additional one. It works fine and if I need the old save data I switch HDDs. However, I would like to play Mass Effect 3 and get the add ons and all ME1 and ME2 datas are on the other HDD. I would like to play ME3 with the same Shepard I built.

What are my options? Do I need to transfer data, if yes How? or Can I start with the old HDD and during the game I switch the HDDs like people do it with memory sticks? I appreciate the help.

You could buy a tranfer cable and do it that way, also.

Thank you all for the answers. From what I understand transfer cables are going to be costly compared to flash disk since I already have. Cloud storage is also unfortunately something I am not familiar with. So I will go with a flash disk. I hope there won't be any formatting issue or it doesn't require a specific brand. I also hope this new DLC for ME3 is something worthwhile.

Download saves to cloud then simply load saves after you install games on new Xbox. KC_Hokie