It’s very easy.Copying and pasting won’t work if you are changing the resolution.
I would do it this way. FIrst go to image>duplicate. That way you have a copy to play around with and won’t be degrading your original when you convert the resolution. Changing your resolution to a lower resolution will result in a lower quality image should you want to later resize it up to 300dpi. You probably know that so you were smart to copy and paste. At least you didn’t damage your original artwork by scaling it down.
Now that you have a duplicate go to image>size and type in your new resolution.Don’t change the width or height dimensions. Photoshop will do that for you automatically. Just be sure that scale styles, constrain proportions and resample image are all checked. Click ok.
Now go back to image.size again and see what your pixel dimensions are. If the image exceeds the threadless limitation you will have to scale down your image. Whichever measurement is too large, be it the width or the length change that to the threadless maximum allowed size. Again, make sure that constrain proportions is checked otherwise you could end up changing the look of your picture. If both measurements are too large change the one that is most over the limitation and just change that.
Now rename your image however you like so you can differentiate from your original higher dpi art. Good luck. Just curious, did you create the image on the computer or did you do it by hand and scan it?
My guess is that he reason you got bad results from cutting and pasting is that you degraded the image by changing the resolution. If you take a 300dpi image and make it 72 you lose a lot of information.(Think of it as similar to converting an audio file to an mp3 which is a lossy formaat) It resizes the image and resamples it so you lose all the detail. If you then size it up again it becomes all pixelated because you have already lost all of the detail The way I am telling you to do it it will retain as much quality as possible while still fitting the guidelines.
It’s a good idea to duplicate a file whenever you’re going to do anything experimental on it. Sure you can undo but this way you can look at the original and the copy side by side and see if you like the changes. Plus you cannot get back information that you lose when you change the resolution. The undo feature cannot recover what has been lost.