Should I have cavity wall insulation installed?
You should always be careful with contractors who seem to be out to make a quick buck. And you should certainly raise your questions with them when you meet them, and make sure you get an answer you are comfortable with. If you are dealing with a government- or utility-subsidized energy efficiency program, you may still have a choice of contractors. And the subsidy program may itself provide some inspection services which can help to keep the contractors in line.

We considered and performed similar energy efficiency upgrades in our home, a 120-year-old wood frame house near Boston. We received bids from two companies, both using the statewide utility subsidy program. The bids included blown cellulose cavity insulation and air sealing. The bids were similarly priced. We ended up going with the company that seemed easier to deal with and more competent in assembling the bid. They in turn used a subcontractor to do the work, and they did good work (probably because they knew to expect a strict inspection afterwards).

The blown cellulose was installed in wall cavities and attic floor. No vapor barrier was added, as this would require tearing out the walls.

The insulation made a noticeable difference in comfort and heating costs the next winter. I agree with you there is some risk of additional damage if moisture gets into the walls; there isn't a nice empty cavity behind the wall to help any leaked water evaporate. This likely means you have to keep a closer eye on your roof and walls, check your attic yearly, and ensure you don't have even a small leak. However, blown-in cellulose insulation is widely used and recommended, so the risk is (probably) small.