The brake lights might have nothing to do with the slow start.

There is a bulb fault relay that tests that current flow is similar to several sets (pairs)of bulbs like head lights and the brake lights. It is red, near the OD relay on the tray in front of the shifter. If the current levels very by such and such a %, then the bulb out indicator lights on the dash. If the relay fails, the indicator light does not light on the dash, and your break lights will be out.

I think I had a post on that, My brake lights went out sept 2004, do a search on the relay number, that will get you some good information.

I took the cover off, and keep tracing solder paths with my ohm meter. Eventiually I found a pin that was connecting, then wastent, So I hit the joint with a small soldering iron and presto, It worked again.

You less time than money you can also buy a new relay, but if the car starts now, and your break lights are out and your fuse is good, and you have power to the fuse for the lights and your break light pedel switch is good, it is the relay.