My question is: why are your arms fully extended? That is, why is that what you want or have a tendency to do? Is your body preferring to be more upright, so you ride with your elbows locked to avoid bending over? If that's the case, Tulip's advice might help a lot.

If you are pretty fit and have neck or back pain when you bend forward to allow a bend in the elbows, you also may need to tweak your fit to accommodate your body better.

If you are already reaching too far, moving your saddle back is probably (imo) going to exacerbate the problem.

Locked elbows is really a recipe for problems.

What happens when you try to ride with bent elbows?

Have you had a bike fit done?

Just some questions before you try radical changes.