There are electrically conductive adhesives. Many more expensive than a second Teensy. I haven't tried them and am curious how well they would work for this application.
Indeed the price is out of this world...altho I found this:
http://www.thinkgeek.com/product/b7...g&matchtype=&gclid=CNbWtdebhL0CFa07MgodU3QAaA
I am ordering a can, just to check how it works. I won't use it on the Teensy, until I find out if it is safe for the long run. It is incredibly cheap...cost more to ship than the product itself! Which is probably the same product that Michael suggested.
If you live in the USA, and are near one of the Radio Shack stores that haven't yet closed, you might try wire glue:
http://www.radioshack.com/product/index.jsp?productId=12925060. Or on ebay:
http://www.ebay.com/itm/Wire-glue-E...881?pt=LH_DefaultDomain_0&hash=item2a37f82d79.
I'm with you in terms of being solder challenged (though I have done some soldering, I'm not that comfortable with it).
I suspect on most projects, people don't use as many analog sensors as you need. The Arduino Uno for instance only has 6 analog pins, and the Teensy gives you 10 analog inputs on the main pins.
I haven't yet gotten a Teensy 3.1, so I forgot that A14 is also easily accessed. Yes, you should be able use that as well on a 3.1 (on the Teensy 3.0, it is the reset pin).
I do live in USA; altho the guy at RS had such a clueless look when I inquired about the product...I ended up finding it at think geek, and ordered a can to try it.
I suck royally at soldering; mostly because I have serious problems with my eyes, and can't even see things with a loupe or the magnify glass...and my hand is far from being steady, which means that in the small areas I have good chance to smear solder all over
Indeed it seems that the most analog input is on the Mega; which is 16 if I recall correctly....today makes little sense to use analog after all, especially when with SPI you can hook as many devices as you want (at the cost of a wire hell); I2C is cleaner but there is the issue with the conflict on the register, if you use the same sensor (I think that in general, 2 is the max number of identical sensor that you can add on any I2C channel).
I went ahead today and woke up with the intention to either succeed or destroy 20 dollars; and I may say that I was decently successful!
I was able to solder A10 and A11, then I went ahead and did A12, A13, A16, A17, A18 and A19. Except a mess caused by me shaking the iron, and dropping a glob of solder on top of the board (god bless solder wick!), I was able to make decent junctions; altho I can't really tell if they are good or not for the long run...looks horrible but overall seems to be able to grab the analog signal from the sensor.
The down side is that I probably burned the LED, since it won't turn on anymore, but the board seems to work (tried few sketch and the serial port is returning the data); if I damaged something else, I have no idea...guess I will find out once I try I2C and try the other mode of the board (in particular I will need USB HID, since the Teensy will act as USB peripheral to drive the sensors).
Attached few pictures to show what I did. I simply tinned and soldered the wires on the plates, then I had to cover the exposed part of the wire with electric tape, since while soldering, I burned also the wire coating, exposing the wire

