A cheap and useful thing to do before getting into a complex modification is to fully clean-out the stock lever-dampers and fill them with new, clean oil. Most folks just top up the dampers, but over the years, there may be all sorts of goo inside them.
Use the proper damper oil. I have heard of some folks using hydraulic oil or motorcycle oil. Some motorcycle fork oils contain an additive to reduce seal leakage, so this may be a good idea for "saving" shocks that are weeping a bit. Whatever you use, do not use motor oil in the dampers.
If you decide to have them rebuilt, I would be very careful about choosing a vendor. In particular, I would not recommend the folks out on Long Island (NY) that run adverts in Hemmings. Their rebuilds are very "spotty"....some good and some bad. Three local foreign repair shops around me will not use them due to quality problems. (I am not mentioning their name on purpose, since that's considered bad form here at BCF....e-mail me if you like to know who they are).
If I were going to stick with the lever dampers, I'd probably buy new ones.
For the record, I have a Spridget and not a "B" and I have converted to tube-style dampers all around (the car is a racer, but also does a slight bit of street driving). On my Spridget, the front conversion made a huge improvement and the rear was worthwhile but less noticable. Some of this improvement involves less "stuttering" (on bumpy turns) and less "fade" of the dampers after a number of hard laps....this may not be such a big deal with a pure-street car. Some tube-shocks are adjustable, which is a neat feature (but usually pretty expensive).
We also have two other Spridgets in the family and both are fine with the stock lever-dampers for street use. On your "B", I'd say you'd be fine for normal street-driving with good-conditon lever dampers.
New A-arm bushings, decent king-pins, decent tie rod ends, a good steering rack and proper clamps on the rear leaf springs will all contribute to good road handling as well. The poly A-arm bushing are fine for street use in my opinion.
If you are curious to see what is involved in a home-brew, tube-shock conversion (for a Spridget....similar concept to converting an MGB), here's mine:
https://npmccabe.tripod.com/spritetubeshock.htm