It could be one the other or both:

A common error for new riders is they try to support their trunk weight with the back rather than the abs. When you ride think about relaxing the back and tightening the abs a bit ( like you're about to start a crunch) so your weight is lying on your abs rather than hanging from your back ( that will also help your breathing).

Also, even the best fit is only going to put you in the ball park. Making adjustments while on a trainer doesn't exactly replicate actual riding. Two possible adjustments come to mind- the "drop" might be too large for your flexibility ( drop = difference between saddle hight/bar hight ) As you ride more and develop better core strength( abs) you're body might become more flexible otherwise you'll need to flip the stem so it raises the bar ( or if already flipped look at getting a new one with more rise ) and/or add some spacers ( if there's room )

Lastly you might be stretched out a little much - try moving the saddle forward a few mm at a time and see if that helps ( if you're hands start to hurt you've moved it to far forward )