Yes, there is a difference. It's subtle, but it's there. The first one means that you will give your phone number through no media other than private message.
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daddyjohnI will ONLY give my new phone number through private message.You will not give your number by any other means, not face to face, not in an email, not via a phone call, not in any way except via private message.
daddyjohnI will give ONLY my new phone number through private message.You will give your