Adding antenna gain to RX and TX sensitivity
Let's say I have an transceiver with 10dBm of TX gain


No, you mean it has a power output of 10 dBm. Gain has got nothing to do with power output capabilities.


  and an RX sensitivity of -10dBm


Yeah that's OK.


  If I then attach an antenna with a gain of -2dBi do I then have an RF
  output power of 8dBm


Effectively, yes you do


  and an RX sensitivity of -8dBm


If the same antenna is at both ends of the link (or it's a transceiver using the antenna for RX and Tx) then yes.


  I'm not clear on if I have to flip the sign when adding to the RX
  sensitivity but that would seem to make sense.


Yes it's a bit confusing but an antenna with -2 dB of gain compared to the (i)sotropic "standard" means you need 2 dB more power into your receiver's antenna because you have degraded the signal with the bad antenna.

And, because you have degraded the power output by 2 dB, your effective power out is 2 dB lower at 8 dBm

In practice, an antenna with loss compared to an isotropic antenna is a pretty crappy antenna and a receiver with a sensitivity of only -10 dBm is also pretty crappy.