To save alot of headaches, is there no way to approach the Police Department, explain the situation and ask if they have it in eps format?
Tips on tracing/redraws:
Fig 1) Best results always come from a complete redraw. The badge is basically symetrical (save the text on it) so you
only have to build one half then duplicate and mirror it.
There is a bit of work to be done in terms of keeping it tidy and attaching the text to the redrawn badge - its in the corel manual (sorry theres alot of nuances in redraws that I can't explain every step)
NOTE: I'm not sure of the legality of a redraw of a badge like this. More than likely you'll need permission.
Fig 2) Good results from tracing need a clean, sharp, high resolution image. Fuzzy edges, low res versions will give you mediocre to bad results when traced. (From Coreldraw you can edit in Photopaint then save the results back into Coreldraw)
This is taking you inbetween tracing and complete redraws but sometimes it helps to convert to a greyscale/black and white image - then you can concentrate on getting the contrast right - but remember this goes hand in hand with the resolution of the image to give you the best possible bitmap edges prior to vectorizing.
Fig3) PowerTrace itself will only get you so far - sometimes an image is just too low resolution to be useable and no amount of editing or cleaning up will get you great results.
Its all dependent on the initial image - the more high res and more detail it captures the better for tracing.
Hope this helps a bit.