My pc tells me that I'm running out of hard drive

You can use either.

You can also use external portable hard disk, you can move all your software and stuff to it.

Better would be installing a new operating system to new hard **** with larger capacity, it will make your system fast.

You can also fit hard disk in your computer PC cabinet, you will just need to set it as "Slave" or secondary hard disk, to do this, setting is available on the hard disk.

you can go for both driver internal as well as external......the thing is what type of data you have stored in 10 gb hdd......if its user data file then you can move it to other external or internal hdd.....if you have installed some heavy software or program then you need to internal hard drive. because you can not copy installed software from one drive to other....for tis you have to install windows on other internal hard drive with bigger size like 160 gb or 250 or 320 gb.....its your wish which one you chose according to your need....