If it's an Android based phone, I can recommend SMS Backup+ which copies SMS to Gmail (it creates a Label in Gmail called SMS) I only use it to backup around 100 messages (as I delete the messages from my phone after doing a backup) so not sure how well it copes with a very large number of messages... expect it will just take a long time. It does a backup of all SMS, don't think you can specify only message conversations between certain contacts/people.
This will backup your text messages, contacts and call history. Just make sure you save it to your microSD and voila. Remove it. When you get your new phone, install Go Backup, insert the card and restore the backup file.
all backed up onto micro sd card to resave later and on laptop. will then be Abe to get them onto new phone when it arrives. with orange insurance you literally hand the casing of one phone and take the new one so no sending from one to other option