No!
The raise of the sea level is not caused by the rain. It is caused by the melting of the ice in Antarctica and Greenland, and small contribution of the expansion of the sea water because it is hotter. Anyway, the current rise is only a few millimeter.
All the rain (and snow) eventually goes to the sea (unless it gets frozen in a glacier) so it doesn't matter if it rains here or there, that doesn't change the sea level.
And remember that there is not a fixed amount of rain, some years there is more global rain, and some years there is less global rain. Essentially, to get rain you need that a big chunk of hot wet air collides with a big chunk of cold air, so the humidity is transformed into clouds and later into rain (and there are more problems, like cloud condensation nuclei, it's not so easy ...) But anyway, how much total rain depends a lot of the circulation details, because humidity my survive in the air for shorter or longer periods of time.
Probably you are cherrypicking some place with drains, but there are other places with more rain and floods. Some of these are normal and caused by the usual random fluctuations of the weather, and others can have a bigger probability due to climate change. It's very difficult to classify them as "normal" or "man-made".