HelloCustomerand welcome to justanswer.com,
No you will not need a new transmission but you will need to have it rebulit. Specifically there is a problem in the fifth gear set. Either the synchros or the actual gear. There is a bulletin from Honda covering this issue,but your vehicle is already out of warranty.
If there is a reputable transmission shop near you they can give you an estimate. Of course if the cost is too high, then you might consider replacing the trans with a good used unit. Keep in mind though if it is used, it could possibly develop the same problem.
I hope this information was helpful. let me know if I can be of more assistance to you.
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