I'm not sure it's the right level for your course, but, here is the advertisement:.
The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss, and Riemann is a story that is often broken into parts - axiomatic geometry, non-Euclidean geometry, and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time.
The presentation is enlivened by historical diversions such as Hugyens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics, Euclid's geometry of space, further properties of cycloids and map projections, and the use of transformations such as the reflections of the Beltrami disk.
If this is not what you want, it at least should give some ideas. It seems to have both axiomatic geometry and the basic differential geometry in a somewhat classical fashion. You might consider basing your course on the Differential Geometry lectures given by Dr. Wildberger [found on youtube, link below. He has some idiosyncrasies that you can clearly ignore, but his approach does not assume a large amount of linear algebra at all.
Notes on Geometry
He does employ matrices later on in the game, but I think with a lecture or two on the use of them, you could be fine. His math history lectures on geometric topics might be good to flesh out the course if the latter part of his differential geometry series goes too far in depth. His course also employs a lot of projective geometry -- which I think is definitely lacking in the curriculum as a whole and would be very worthwhile!
Here is the link to his playlist: Diff Geo Wildberger. I cannot answer your question. But since no one else answered, let me reply with another question: Would it be feasible to base an undergraduate geometry class on Thurston's great book? William P. Three-Dimensional Geometry and Topology: Volume 1.
Notes on geometry / Elmer G. Rees. - Version details - Trove
Edited by Silvio Levy. Princeton Univ Press link. Added to basket. Drawing Geometry. Jon Allen. Algebraic Topology. Allen Hatcher. Mathematics and Its History. John Stillwell. Paul Lockhart.
Introduction to Topology. Bert Mendelson. Fractals: A Very Short Introduction. Kenneth Falconer. Modern Graph Theory. Bela Bollobas. Benjamin Bold. An Introduction to Manifolds. Loring W. The Four Pillars of Geometry.
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Introducing Fractals. On quantum cohomology rings of partial flag varieties. Duke Math. MR d [C1] I. Algebraic surfaces and holomorphic vector bundles. Springer-Verlag, New York, MR 99c [Ful] W. A Series of Modern Surveys in Mathematics.
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Related Notes on Geometry (Universitext)
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