If I append "1", "2", "3" etc to the original series of rolls (seed) - sha256sum(3315135445366124436162446626244624624266466466622442224426424444624626624246264 4644646626242462622241) .. does this constitute a "secure" method for generating a series of addresses?
What I would personally do is tack on a letter, say "x" and then use a 3 digit number with leading zeros.
For example:
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x001
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x002
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x003
...
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x998
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x999
3315135445366124436162446626244624624266466466622442224426424444624626624246264
464464662624246262224x000
Would give you a thousand addresses.
If it's just a few you need, I'd just roll another 100 times for every address. There is no need to roll more than that for each address, you'll be wasting rolls.
The dice rolls are secure as long as no one else is in the room, and no one else is watching you roll or can see the dice. The computer you use is secure as long as it is offline and is properly wiped or erased or encrypted or everything is run in RAM then completely powered off.
To "properly" roll the dice means you should throw it across the room, and record only the ones that bounce back from the opposite wall. (Or roll it on something similar to a craps table, where it has to bounce from the other side.)
If you're lazy and already have 100 dice, just throw them all at once against the wall and take a RAW picture of everything using a 24 megapixel DSLR camera at the highest ISO setting, then hash the 15 megabyte file for your new private key. Then delete the picture and format the memory card.