First, get a second opinion and not at a 'transmission specialist'. Rather than getting it 'rebuilt', which is quite expensive, call Erievovo in NY state and ask for a used one. I would also ask if a 2004 tranny could be placed in your 2003, the 04 is better built.

The following from IPD which discusses the B4 servo cover:

Product Description
It's ironic how this inexpensive part can save someone from a costly transmission replacement.

The early AW55-50/51SN five speed automatic transmissions have an issue with shift “flares” and other odd symptoms related to shifting in and out of 3rd gear. Usually, this is exhibited by a “missed” shift (somtimes called "shift flare") from 2nd to 3rd gear on a random basis.

This B4 servo cover update can solve these symptoms in many cases. Other times, it may take a software update or the B4 problem has gone on too long and the transmission is physically damaged. It is possible for transmission damage to occur when the transmission hooks hard up after a flare condition.

Volvo has changed the design of the B4 servo cover multiple times and there are numerous Volvo service bulletins regarding it. The latest design is stock on later model years.

We highly recommend a transmission software upgrade prior to or in conjunction with the B4 servo cover update!

The software for these electronically controlled tramissions has gone though many version changes over the years and has become much more refined. A software upgrade can significatly reduce neutral to drive delays and shift flare. If your car still rolls backwards when in drive while facing up hill on a slight slope, then you most likely need a software upgrade. The neutral-at- a-stop while in Drive "feature" was removed with later software versions because it can cause damage when the transmission engages hard. Software upgrades must be done with a Volvo specialty shop using Volvo's VIDA application.

The B4 Servo issues usually don't affect 2004 and newer models, but some of these cars still have a variety of shifting issues that can be solved in many cases by a transmission software upgrade.
--
My name is Klaus and I am a V ♂ lv ♂ holic