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it really depends more on the lens you have. a crop sensor will get you "closer" because it gives you a higher 35mm equiv. focal length, but in reality it gets you closer because it's a crop sensor. which is the same as taking an image on a full frame sensor and cropping it down to get in closer.
pick a good macro lens. then find a camera in your budget with your requirements that will fit that macro lens mount. it doesn't matter if it's full frame or crop sensor. and pick up some cheap extension tubes.
Without a doubt the larger the sensor the less noise but few options in lenses for macro are available. I would recommend a full frame (FX) sensor as a good balance between available lenses for macro and sensor size. A key thing to bear in mind is that a longer focal length macro lens has more working distance from the macro subject and therefore has better opportunities for lighting and actually getting the shot if the subject is alive and nervous. About a 100mm is a good point to start though I use a 150mm myself (on a full frame camera).
I have both DX and FX cameras and strongly recommend full frame if financially possible. DX can produce great results but great care is needed to get a great composition in camera to maximise its available quality.