1997 Accord electrical probs?

1997 Accord 4 cyl. 265,000 miles -- manual transmission

Background problem: sometimes won't start in damp/wet weather unless I blowdry the distributor cap, after removing plugs. Otherwise, no problems starting.

Current situation:
After a short rain, I moved the car into garage -- it started up no problem, so I thought nothing of it. Next day, car wouldn't start -- just got clicking noise when I'd turn the key. Turns out after testing, the battery wasn't holding a charge all of a sudden.

I replaced the battery with new one. The car fired up immediately. After about 75 minutes of driving (30 min, short break, then 45 minutes), I noticed that with the headlights turned on, the speedometer & odometer would fall to 0 (not be engaged) so I couldn't tell how fast I was driving. Turn off the lights and the speedometer/odometer come back to life.

However, if I kept driving, eventually all the dashboard lights dim, speedometer/odometer falls to 0, and then the final thing is the car starts to jerk as if to stall.

I'm guessing either I have a bad battery, a bad alternator or something else I'm not thinking of. Any ideas how to localize the problem?

Thanks in advance.