The volitional form at the end of a sentence doesn’t always refer to a joint action by the speaker and the listener. Under certain circumstances it refers to an action by the speaker alone. One example is when you offer to do something for someone. Usually, you first ask whether that is indeed what the other person wants by turning it into a question by adding か. This is the third and last pattern in this answer.
The version without か, unless it is said in a monologue, is kind of like answering to your own such question before you ask it. It is like letting the listener know what you are going to do for them. The action is most likely something from which they benefit. It could be part of joint efforts towards a common goal.
Without volitional forms, your sentences would sound more like unilateral declarations that leave absolutely no room for other people’s involvement in decision making. Depending on the context, you could even sound as if you are either refusing their involvement or telling them you are having to do whatever it is on your own because they are uncooperative.
I would summarize the difference as one in the degree of involvement on the listener’s part. I feel a similar nuance between “Let me …” and “I will …” in English.
Then, let me help you, too.
Then, I will help you, too.
Let me see that letter (on our behalf).
I will see that letter (as opposed to anyone else).
Let me (be the one to) vouch for Sasuke.
I will vouch for Sasuke.