A MSP430 will take a maximum of around 8-15 mA in active mode unless you're using a RF MCU or have heavy current sourcing from the I/Os. If you're programming the thing to sleep most of the time in one of the LPM modes (as you should for a battery powered application and desire long run times), most of the time the MSP430 will be consuming microamps of current sleeping. If you limit the active mode clock frequency to 1MHz or even less (maybe even 12kHz using the VLO or 32kHz) your active mode power consumption will be sub-1mA.
Use sleep mode and very low duty cycle active mode runtimes. Use low clock frequencies and perhaps low supply voltages to promote battery usage efficiency.
Size the battery mAh capacity to handle the runtime you want taking into account battery self discharge / leakage, the sleep mode vs active mode power consumption of the unit over time (mAh averaged over a day, week, month, year). Since you're using AAs you know what mAh capacity and self discharge rate to expect. You can use a 3S arrangement of AAs to get from 2.7V to 4.8V or 2S to get 1.8V to 3.2V. In the 3S case you can use a very low quiescent current LDO (possibly one with pass through capability when the battery voltage has declined to within the MSP430's operational tolerances) to power the MSP430 and still have high battery usage efficiency. For the 2S or 3S cases you'll want to use some kind of a boost converter to generate the 10V for the load cell. If the current requirement when "ON" is low enough a charge pump type may suffice, or you can use a small switcher. Look for a unit with a very low shutdown mode current (in the sub-10uA range) and an output isolated from the input during shutdown if you can use the MSP430 to control the boost converter's wake up from shutdown mode only when the load cell needs to be powered up.
Check out the TI WEBENCH and Switcherpro tools on their web site to assist with optimum part selection. There are LDOs specifically tailored for use with the MSP430 which have very low quiescent current values and which can provide up to 75mA of current for the MCU.
Edit : Adding specific part information.
The TPS780x and TPS782x are low Iq LDOs specifically recommended for use with the MSP430, and there are application notes detailing their benefits in those solutions. The TPS78227 or TPS78228 will give a regulated output down to 2.7 or 2.8V respectively. There are higher and lower voltage members of the family. The TPS61041 will boost battery voltage up to the level needed for your sensor and is a reasonably compact solution. Other models with integrated LDOs and/or higher frequency operation are available if you should wish further noise reduction.