You can definitely do a great Liver Flush without epsom. I really like the apple juice protocol at sensiblehealth.com.
However, epsom isn't something that should cause constipation; the reverse is true! Something else is causing this problem--I would guess something to do with the excess bile you are flushing, though again I would think it would cause the opposite reaction, diarrhea. What are you eating before and after the flush? Are you eating very low/no fat for a day or two before, and again after? Very clean diet? Constipation is a very odd thing post flush.
I give Liver Flushing full credit for the complete, sudden change for the better in the condition of my skin. However, I had done a LOT of serious colon cleansing previously, AND I was eating a very clean, healthy diet, drinking lots and lots of water, getting regular exercise etc. Of course there is much more to clear skin than just doing the liver flush. My own program was long and complicated, and the benefits extend far beyond clear skin.
Around 30% of our body's wastes are excreted through the skin. This is why I am fully convinced my own cure is due to the major cleanup of my personal toxic waste dump! Clean up your colon, clean up your filters (liver/gallbladder, kidneys), clean up your diet, drink lots of water and get regular exercise. Easier said than done, right?
I still believe that YOU CAN DO IT! If you are experiencing trouble with your skin, there is an internal cause or causes. Find the cause and correct it and you are on your way!