Check your firing order 1342
Starter's located below the distributor, mounted on top of the tranny.
You can find your 1988 Acura Integra starter on the right hand side of the engine. You can follow the positive battery cable directly to the starter.
Take it in to your nearest Autozone and have the starter checked. Before having to push or tow your car, make sure both leads are securely attached to the battery and other wiring; from the alternator to the distributor and distributor cap to spark plugs etc.
check the starter
remove air intake battery and battery plat form right there under air intake on transmission
The starter is located on the back of the engine, near the bottom. You can follow the positive cable from the battery directly to the starter.
I am not entirely sure since i have a 1994 integra ls, BUT.. you should probably check/test these on you're car> starter, igniter (Ignition), your alternator. Try checking terminals, one of them could be ruined on you're starter due to the rain.
Bad starter solenoid? Loose or corroded battery terminals? Weak battery? Bad neutral switch?
Is it charging, if not have you checked wiring to alternator.
a, the battery condition should be checked first to make sure that its capable of turning over the starter, a weak battery will cause the majority of starter problems, if the battery is fine then its more than likley to be a problem with the actual starter or poor connections on the starter, i have only ever had 1 with a wiring problem, as for the timing out it will not cause slow cranking unless its well out causing piston to valve contact this is only possible if the belt has slipped or it was timed up wrong on refitting a new belt so b is highley unlikley but not impossable. Checking the timing went away with the distributor.
Follow the + battery cable should connect to the starter relay (solenoid)
the starter? the battery? Distributor? idk give more information... does it click?