Nope, the Arabian peninsula is surrounded by water and it's a desert. So Is all of North Africa and Australia. The rule you mention is an exception around the tropics. Being next to the sea do not guarantee having rain. I think the confusion comes form the fact that most people will understand the air movement in 2 dimensions only. But in reality, it's in 3 dimensions. The descending air over Arabia is dry because it comes form a higher altitude, possibly around 10 km.
The other thing to consider is why does it rain? Having a lot of water in suspension is not enough to create rain. It can stay in the air for a long time. In order to fall, it needs a confrontation with cold air. As the air cools off, it cannot contain all the water. Becoming saturated, it start to rain. The more drastic the temperature gradient, the more violent the reaction is.
Look at you windows: If you live in the north you probably noticed water condensation during winter. That's the same principle as rain.
So, I believe the climate on this part of your world should be similar to Africa (turned upside down). Tropical rainforests near the equator with an increasingly dry season as we move toward the tropics. Large areas would be deserts around your seas. And the southern portion as a North Africa like climate with the southernmost area similar to Southern and Western Europe.
That being said, most of your seas either fall into a steppe climate or a desert climate. This means a lot of evaporation. I take lake Nasser (an artificial lake on the Nile) in Egypt for comparison: it contains about 132 cube km of water and apparently he lose 10-16 cube km each year due to evaporation. I means that without external water addition, it would lose 10% of his volume each year. This seems like huge numbers but lake Tchad / Chad also as a high evaporation and is a seasonal lake, the lake can almost disappear during the dry summer.
Based on the Sahara, we can estimate that this land would need between 3m and 6m of annual precipitation in order to be considered as humid as the Mediterranean climates. It's half of this if it's under maritime influence of near a large lake.
Most of you lake only have marginal inputs. With sufficient time, about half of the lakes in the middle will disappear and another bunch will almost disappear in the dry season. Of course that depend, if the water flow toward the desert, it's possible to have lakes there. Just like lake Nasser that would be dried quickly without external water sources.
This brings me to question how these seas/lakes appeared in the first place. Was it an ocean that was trapped between 2 landmasses? It seems improbable because it take thousands, millions of years to complete and by that time, the seas would have dried out.
Option 2 would be that a drastic change in the climate occurred in the last centuries perhaps. Or change in the axial tilt. The region used to receive more rain, enough to feed these large lakes with fresh water.
Or maybe it's a combination of the two.
Lastly, their impact of the climate could be significant. I doubt it would rain much but the relative humidity would be higher than in the central Sahara. The most important difference would be that the temperature would tend to be lower during the day when near the water but also warmer during the nights, compared to areas that are far from the water. This is just because water takes more time to change temperature.