Old-fashioned women's style bikes have the top tube running parallel or downwards next to the downtube.

I've wondered, how does this affect the riding/function? I imagine it's less strong, being not a triangle, but that is rarely a real concern. Is there more vibration in the front because it's connected on a narrower set of tubes?

I'm sure there's a difference in feel or function, because otherwise it'd seem that a sloping top tube would make more sense for all bikes practically, because then anybody could use almost any bike and not worry about standover height.

(My inseam is ~ 28 which is why I'm so concerned with bike sizes)

Difference in function: it allows you to ride in a skirt or kilt.

Difference in feel: minimal. The vast majority of these bikes were built with heavier-gauge tubing, so any difference in "feel" would be masked by the different tubing gauge. Besides, any differences in frame design or tubing is much smaller than the differences made by changing tire pressure.

I think the women's frames that you are describing may be weaker because the forward triangle left is open due to the lowered top tube. This can place additional stress on the seat tube and stronger tubes may be required possibly making the bike heavier. This could mean the frame won't last as long due to the stress.

The double triangle/ diamond frame is tried and true which is why it is the defacto standard for most bikes and even some women's models. If you look at some mid-range or high end women's bikes like Trek 7.5FX WSD, they still retain the double triangle frame but come in small frame sizes like 13 and 15 with stand over heights as low as 26inches which might work out for you.

Another option is a mixte frame which is similar to the women's frames you are describing but uses two smaller diameter tubes that run straight back to the rear wheel. These are sort of a niche market though, so any new bike you find with this type of frame might be out of your budget.