To clarify "unlimited texting" or SMS is for just text type messages (words). Videos and Pictures are excluded. Blackberry's BBM platform DOES support rich media.
You need to send your pictures via email if your phone has a data package. I believe that phone has a 5 megapixel camera. If you are not careful, that can quickly eat up a data plan.
And the other phone was????
If it was a Dumb Phone or Feature phone, it may have used the older technology to push these items through that network. Smart phones almost all use the data network to move photos, etc. Voice and SMS are on the voice network, and all data is on the data network.
You can add data to your prepaid phone, but be aware that a 5 - 8 megapixel photo uses a lot more data than the little photos you sent on the old LG.