How can an organization increase the accuracy of the budgeting process?

By doing it more accurately.  Instead of just taking the total costs
of a department and estimating that if sales increase by 20% the
department's cost will also, one should look at the individual
expenses of the department.  Maybe floor space and related expenses
won't change at all, but postage will increase much more, because the
anticipated increase in sales will require even more mailings to
prospective customers that don't result in sales.  Tracking expenses
on individual expenses for previous accounting periods will allow
greater insight to budgeting for the future.

Ditto for sales:  estimate for each product or group of products.

In addition :

By rigidly auditing what is posted against an account, and making sure
people know what is going on.

I once raided the PR budget to increase a print run ...

Actually people are generally rather good at budgeting, but there are
seldom advantages for coming in under budget (like it gets reduced
next year) and it is always useful having a slush fund.