The reason many people recommend to send them separately is because some low-level IRS clerk may staple them all together as one return, or pages may get shuffled or lost. There's less chance of something getting sidetracked at the IRS if sent separately.
There's not much advantage to sending them by a private delivery service (FedEx, UPS, etc.) since you are not under a tight deadline, and since you are due a refund rather than paying taxes due. Private delivery service would more commonly be used at the last minute when an IRS deadline is imminent in the next couple of days.
Regardless of how you send them, it's still going to take the IRS about 6 weeks to normally process a paper return, so expedited delivery of 2-3 days is not going to make much difference. By the way, that 6 week estimate is if everything processes normally. The IRS can always pull a return for additional review and processing, which can take longer.
It's up to you to decide how much money you want to spend sending them, and which method you trust the most. With 6 separate returns to send, it can add up quickly.
I normally use US postal mail--specifically certified mail with return receipt--addressed to the designated IRS postal mailing address for your state. The state returns go to the state tax agency. That method gets better handling by the Post Office and special attention than regular US mail. And it includes tracking, proof of mailing date, etc. It will still cost a fair amount for 6 separate packets, but I suspect less than FedEx/UPS, etc. You'll have to check.
