How can I get the objects not to resize when the user resizes the browser?
Sure you can! If I understand your question correctly, all you have to do is to change the height and width tags to a fixed value (and I put the flash file into a table too). I put two weatherflash instances on the same page here in two different sizes (to illustrate the idea): http://www.robuck.net/weather/temp2.html
If you can't figure it out, just look at the source code....in one case it's the default size, 800 x 487 and in the smaller one its 400 x 243.