A floor plan would help.

I have seen this situation before in other homes and usually it is designed that way because of a lack of space.

Ideally the stairs would lead to a landing at floor level with a door in either direction leading to the kitchen and the living room.

The only way to tell is for me to see a floor plan.........

The landing is between the kitchen on left and living room on right. There are three steps on either side of the landing leading right into both rooms. The house is in great shape needing mostly cosmetic fixes but this is really throwing me for a loop!

What do you find awkward? Is it moving between the 2 rooms having to go up and down?

If you want to know your options, you might have a local architect consult with you at the house. If there is a solution she would have suggestions and should also be able to give you an idea of costs of construction...