Low Pressure It runs, and sprays water, but doesn't have near the pressure it had. Its also losing water out a weep hole on the water pump. Manual says check the screens, but I took it apart and didn't find any. Any suggestions?

is it losing water? if not is your fan kicking on? if losing water, place piece of cardboard under veh at night to help pinpoint where water is leaking. if not losing water and fan does'nt come on check fan relay. if loses water and don't see leak over night on cardboard, then water pump is leaking thru weep hole but only after heats up and pressure is on need to replace water pump.

Usually, there are spaces between the pump and head portion of the pump. This space is not sealed and will allow water leaking past the seals to come out. There may be a seal kit available, however I have had better luck just changing out the whole pump. There are many available on ebay. Search for one that will use same mount and hoses. Once the seals are gone, the pump will never be the same (may nickle and dime you the death).

REPLACE THE THERMOSTAT AND RADIATOR PRESSURE CAP.IF COOLANT LOW, ADD MORE COOLANT THROUGH COOLANT OVERFLOW UNTIL COOLANT LEVEL IN THE COOLANT OVERFLOW STAY AT FULL COLD MARK.REASON POUR COOLANT THROUGH COOLANT OVERFLOW MOST GM CARS DONT HAVE RADIATOR CAP.WHEN YOU GET COOLANT LEVEL OKAY, CRANK CAR LET IT IDLE UNTIL ENGINE GET HOT BUILD UP PRESSURE THEN TURN OFF CAR.CHECK FOR COOLANTS LEAKS AT THE RADIATOR HOSES,CHECK FOR LEAKS AT THE RADIATOR, CHECK FOR LEAKS AT WATER PUMP WEEP HOLE IF YOU SEE COOLANT LEAKING FROM WEEP HOLE REPLACE WATER PUMP. IF CAR OVERHEATING WHILE IDLING YOUR COOLING FANS NOT WORKING CHECK COOLING FANS FUSE AND RELAYS.IF FUSE AND RELAYS OKAY THE ENGINECOOLANT TEMPERATURE SENSOR BAD.

usaully a hole near the shaft of a water pump is a weep hole and if water or antifreeze is coming out, your water pump seal is bad and pump needs replaced.

Your pump seal is leaking and is causing you to lose pressure and fluid. Do not drive it! You will burn up the clutches as the pressure with be low and will not fully apply the clutches. Transmision must be pulled to replace.

No it is not a cracked head, your water pump has a pressure hole to give you an indication that the pump is getting there. when the motor is running the pressure builds up and squirts the water all over, resulting in misleading you.

Can you see where the antifreeze may be coming from. it could be the weep hole in the water pump letting the water out, this is "saying the water pump is bad " or it could be a simple problem as the coolant cap isn't holding the pressure and needs to be replaced. This will cause a car to over heat and lose coolant.

CHECK WATER PUMP WEEP HOLE IF ITS LEAKING.THEN PUMPS SHAFT BEARINGS FAILING MEANING YOU NEED A NEW WATER PUMP.IF WATER PUMP OKAY.I BE LOOKING AT THE THERMOSTAT AND RADIATOR PRESSURE CAP.

Yes the water pump does run of the timing belt system.
Directly from the service manual:

The 3.5L water pump has a die cast aluminum
housing and a plastic swept vane impeller. It bolts
directly to the right rear timing belt cover using an
O-ring for sealing. The water pump is
driven by the engine timing belt.

REMOVAL
The water pump on all models can be replaced
without discharging the air conditioning system.
WARNING: DO NOT REMOVE PRESSURE CAP
WITH THE SYSTEM HOT AND UNDER PRESSURE
BECAUSE SERIOUS BURNS FROM COOLANT CAN
OCCUR.

NOTE: It is normal for the water pump to weep a
small amount of coolant from the weep hole (black
stain on water pump body). Do not replace the
water pump if this condition exists. Replace the
water pump if a heavy deposit or a steady flow of
engine coolant is evident on water pump body from
the weep hole (shaft seal failure). Be sure to perform
a thorough analysis before replacing water
pump.

(1) Drain cooling system (Refer to 7 - COOLING -
STANDARD PROCEDURE).

(2) Remove accessory drive belts (Refer to 7 -
COOLING/ACCESSORY DRIVE/DRIVE BELTS -
REMOVAL).

NOTE: The water pump is driven by the timing belt.

(3) Remove engine timing belt (Refer to 9 -
ENGINE/VALVE TIMING/TIMING BELT AND
SPROCKETS - REMOVAL).

(4) Remove water pump mounting bolts (Fig. 31).
Note position of longer bolt for proper re-installation.

(5) Remove water pump body from engine

Here is link to get a factory service manual relatively cheap.
2004 Chrysler 300M Service Manual Covers every serviceable part.

low coolant level , restriction, bad water pump, or stuck thermostat.