2003 Dyna Low Rider. Carburated Engine runs smooth at 5000 ft. altitude, but cuts out when riding at 1000 ft.

Carb engine I would guess? Probably running lean at lower altitude since air is denser .

Not sure your bike, but carbureted engines need to be leaned out at higher altitudes due to the thinner air.
And set back when back at lower altitudes for maximum performance.
I know I haven't had problems at 8,000 ft. but also wasn't checking performance?
If Altitude is a problem, carbs are what needs to be dealt with.
Check your air filter.
I was thinking you meant 10000 ft. If 1000 ft, Carbs need more fuel.