99 ford escort starting hard?

I have a 99 ford escort that is starting hard. I have to let the fuel pump kick on at least twice before it will start. Once it's started it runs pretty good.
The first thing I did was change the fuel filter, didn't help.
The next thing was take it to the garage and they said it needed a fuel pump because it was losing pressure… They wanted $500 so I changed the fuel pump myself, seem to help for 1 day then back to starting hard again.
The check engine light has not come on.
The fuel pump assembly had to little screen filters that looked dirty, but I couldn't remove them without breaking them so I didn't change them.
Anyone have any idea what else could be wrong?

As far as I can recall there's a check/non return valve under the screen filter

I would do a fuel pressure test to diagnose this problem.

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