You would not want to trust blindly (especially after what had happened with your ex's betrayal)
It seems a bit too much to text daily a female friend even if she lives out of state. It is not unreasonable fo you to explain to him your need for reassurance. It is also respectful for him to at least tell you what is it that they are conversing/texting about so much.
Writing "love you and miss you" to each other is totally inappropriate when he is in a relationship with you. If there was nothing to hide, why the secrecy on his part?
He may be having an" emotional affair" with her seeking her compliments to boost his self esteem, etc. If you continue the relationship, you would want to figure out if this is how you want things to go on. Today he texts her, tomorrow he may start texting someone who lives in your town down the street. Compromises in a relationships are ok as long as you do not compromise your happiness and well being. Figure out if this is a serious committed guy or not. Your love for him won't change him. He has to do that himself. Sibmissiveness is not healthy for any relationship (it has to be equal)