OP, I think that this is the best and safest approach, also pretty well written.
If there's a big amount there, this is what you should do.
And if you're not sure, you can ask, and ask, and ask again, for each and every step, until you can do this.

I will add that:
* you may want to not send the entire amount to exchanges (if this is the reason for touching those coins)
* you may want to make sure the destination wallet is safe (maybe hardware wallet, double checked that the recovery seed is indeed correctly written and works) if you touch the coins in order to send them away