BLUF: In your situation, you can only do it through reenlistment.
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Detailed answer:
Overall, you should become familiar with AR 614-200* which covers a wide variety of personnel/administrative actions for enlisted Soldiers, including reclassification (Section III).

'How' depends if you're in your reenlistment window or not. If so, then you can only change it though reenlistment.

If not, then you work with your retention NCO to see if you're eligible to voluntarily reclassify and to process the request. The key question is if you're one of the 'square pegs' they are looking to take out of a 'round hole' and put into a 'square hole'.

What do I mean by 'square peg' - HRC periodically sends out a MILPER message showing the IN/OUT Calls* - these are the MOSes that are eligible to reclassify voluntarily - you must be eligible for your rank to be released and they must be looking for your rank in the MOS you want to go to.

Looking specifically (see picture) at an E5 68W going to an E5 68V, the message says that while 68Vs are accepting non-promotable E5s, 68Ws are not releasing them (just E4s and below).

I'll defer to SFC (Join to see) if I have anything wrong here as he's much more familiar with the enlisted personnel system than I am.
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* https://armypubs.army.mil/epubs/DR_pubs/DR_a/pdf/web/ARN14314_AR614-200_FINAL.pdf
* Latest IN/OUT call MILPER - https://www.hrc.army.mil/milper/22-379

I did mine via a re enlistment. They used to have the B.E.A.R. (Bonus Extension And Reenlistment) program, if your MOS was ovverstrength and the gaining MOS was understrength is was doable. I left 11B and went (91V now 68V). Respiratory was very demanding but rewarding. Promotions are better if you remain a medic, but for life after the Army, Respiratory is a great way to go.

68V requires a transcript evaluation. If you go on the ATRRS Course Catalog you can see what they are. There's an email you have to send your transcripts to for evaluation. I'm not sure why you're Career Counselor hasn't given you an accurate answer, it's a pretty clear and simple process. If your Battalion Career Counselor can't accurately answer it, then go to your Brigade Senior Career Counselor who has experience with this.