I'm getting calls on my land line from people that say that I've called them when I haven't.

Someone is using your number in their caller ID
Someone has tap into your phone
Or there is a switching issue at the mail phone terminal and phone is showing up as yours

Is my only option changing my number and/or carrier?

Get your number unlisted is one option.

They are likely lying Pieces of poop(cleaned up).
I get many calls saying I spoke with them last month and were following up to sell me crap.
I am registered with Donotcall.gov, But Bullters are immune from the law.
Anyone else in the household using the phone?
You could get a call manager, Like a caller ID on Steroids, It log all incomming and outgoing calls. The outgoing calls will show what time and the number dialed.
It may show the time that a neighbor may hook on to your line and making calls? If as Father Chuck mentions.
A recorder on the line would provide a voice making the calls.

Telemarketers are desperate. Land lines are going the way of the dinosaur. So marketers say anything. I moved to a rural area with no cell coverage, and just got a land line, and all I get are calls from people I don't know, mostly collectors of VERY old debts from when someone else had my phone number. They don't believe that I am not or don't know the person they are trying to reach. Hang up on them!