How to say “similarly" as a transition?
Actually, there is no different meaning between 同样的 and 同样 when we use them to connect two sentences.

However, in my experience, 同样的 is used to connect two short sentences, like your first example. Tom swims well, 同样的, John can swim with three styles.

同样 is used start a new sentence. Tom is not rich, but he is really a nice guy. 同样, even John is disabled, but he try his best to take care of himself and help more people.

In speaking Chinese, you can ignore this. Because the pronunciation of 的 in 同样的 can speak like the “d” in yard.