What part(s) can I use to connect modules at a 90 degree angle? I'm looking for brainstorming! All ideas welcome.

I have a design I'm working on which:


is made of a bunch of identical modules fitting into a single framework
has to allow the modules to "plug in" to the framework (be removable)


I'm designing the framework.

The problem is that in some places the modules meet at an internal corner in a way I can't connect: two modules are connected to the same framework brick (which would be in the space next to both of them).

Here's pictures of two modules meeting, without the framework, which would be in the space defined by the corner:




Is there a piece or assembly that matches either of these scenarios?


one suggestion has been Bricklink Part 10197 the "Technic, Axle and Pin Connector Hub with 2 Axle"
however, with this part, since the pins are mid-stud, the axles reach half a brick in.


Questions:


any part suggestions for either of these configurations?
any redesign suggestions?  Does anyone have a favorite combination for an internal corner like this?


Possible solutions so far:


use axles and the above Bricklink Part 10197
use axles and an as-yet unknown part which is a better fit
use pins and an as-yet unknown part which fits at all
use pins and move the pin hole to a single brick Technic brick, which has a hole aligned with the stud
move all the rightmost connectors slightly up and all the leftmost connectors slightly down (this is the least desirable for aesthetic reasons)


See also the question on this site, "What part(s) can I use to connect technic liftarms at a 90 degree angle?"
Here are all possibilities I can think of (most of them already listed in your question):


Use part 10197 Technic, Axle and Pin Connector Hub with 2 Axles together with 2x 4265c Technic Bush 1/2 Smooth, one on each axle end. This will keep it snug and prevents it from moving, but the axles are, as you observed, only halfway inserted into the bricks.





Similar solution, but use 22961 Technic, Axle and Pin Connector Hub with 1 Axle combined with 4265c Technic Bush 1/2 Smooth, and place 18651 Technic, Axle Pin 3L with Friction Ridges Lengthwise and 2L Axle in the hole, together with another 4265c Technic Bush 1/2 Smooth. This combination will have one end solidly inserted (actually sticking out 1/2 a brick) and the other end is halfway in.





Exchange the "standard" Technic 1x2 brick for a 6541 Technic, Brick 1 x 1 with Hole, or even better, a 32000 Technic, Brick 1 x 2 with Holes. Then use a 44809 Technic, Pin Connector Perpendicular 2 x 2 Bent and two normal friction pins to make the connection. This should be rock solid.




The solution would look like this: 





BEGIN EDIT 2016/03/04:


I recently came across this piece that would be IDEAL for your original purpose: Part 6553 Technic Pole Reverser Handle. 




This piece puts the axle it can accept in the opening at a 1/2 stud distance from the brick it connects to, which would work perfectly in your original design. Combine this with any axle and you get the following solution:



This solution has the advantage that you can take it apart much easier than solution 3, which is more of a "permanent attachment" option. The axle connected to 6653 is 1.5 long, so it does stick 1/2 stud out from the back of the +-shaped Technic brick. You will need to account for this in your design.

END EDIT 2016/03/04