Speaking from a woman's point of view I can see why Liz over reacted to Beth. Liz is 42 and Beth is 23 and by your own admission they are similar in appearance. As women we are told over and over again that being older is negative and that your man will want to replace you with someone younger. Since things haven't progressed beyond a friendship with Beth it should probably stay that way for the time being until Liz has a chance to process why she is feeling the way she is about you being involved with Beth. Try to meet some new people and see how she reacts to others. Take things slow and keep talking about issues as they come up.
Everything will be ok in the end. If it's not ok it's not the end.