I realise this feature was removed, however I have still been using it. Today I noticed that messages are no longer syncing and messages data has been removed from iCloud. Did Apple turn the feature off recently?
I had used it in an earlier beta and it did work well on a restore from new. All of the picture messages went to the cloud, creating space. But now, when I click on "download" it won't show the message on the cloud. I hope when they do bring it back, those old picture messages are available.
There are ways to extract them from an iTunes backup but (a) it's not easy and requires some 3rd-party app plus (b) you say without a computer.
Without just selecting every single one and forwarding them to someone who has more space than you on their phone/iPad, I'd say no, there's no easy way. Treat iMessages and texts as quick little phone calls. Let them fade from your memory and get deleted - you don't need them ;p