Is the flow of [name_f]Elizabeth[/name_f] [name_f]Ruth[/name_f] horrible or is it passable?
I wouldn’t say it’s absolutely terrible, but the -th ending for both makes it sound very choppy.
I agree with [name_f]Emily[/name_f], it’s not the rhythm just the similar endings. [name_f]Elizabeth[/name_f] [name_u]Rue[/name_u] would be a fantastic alternative. (And if you are honoring someone with [name_f]Ruth[/name_f], I think [name_u]Rue[/name_u] would still do the job as well) 
YES! [name_f]Elizabeth[/name_f] [name_u]Rue[/name_u] is adorable!! 
I agree with @dramagrl19 about two “th” endings. Why not go with two middle names to break it up i.e. [name_f]Elizabeth[/name_f] [name_f]Josephine[/name_f] [name_f]Ruth[/name_f]?
I agree–the “th” endings make it very lispy. It’s fine as a combo, though. [name_f]Elizabeth[/name_f] [name_u]Rue[/name_u] is adorable, but I find myself drawn to [name_f]Elisabeth[/name_f] [name_u]Ruby[/name_u], personally. A bit more feminine, and more spunky, too. Or even [name_f]Ruth[/name_f] [name_f]Elizabeth[/name_f]?
Given that this is all theoretical I would say it’s not a great combo and you should keep looking. If you were older and were going to be giving birth to what you knew would be your only child, I would say it’s that not that bad and it’s not. But it’s also not great. My friend had a lisp and found ths hard to say so [name_f]Elizabeth[/name_f] [name_f]Ruth[/name_f] could be really hard to say out loud.
My solution would be to find other combos that work with either [name_f]Elizabeth[/name_f] or [name_f]Ruth[/name_f].
Good luck!
If [name_f]Ruth[/name_f] is honouring someone in your family, then it’s not a deal-breaker. Family signifance trumps flow.
If [name_f]Ruth[/name_f] is just a name you stuck in the middle, I would advise that you keep looking for an option with no repetitive sound with [name_f]Elizabeth[/name_f] (th).
I like the suggestion of adding a second middle name, especially [name_f]Elizabeth[/name_f] [name_f]Josephine[/name_f] [name_f]Ruth[/name_f]!