This is 三河弁｛みかわべん｝, a dialect used in parts of 愛知県｛あいちけん｝.
Conjugating with りん is a sort of "soft" imperative, suggestive rather than commanding.
As for your sentences, the parts with the りん therefore translate to the following:
You (亜也) are good at English, so you should master it completely.
(It is good to) hold the hand rail while going up.
It's definitely dialectal, but according to this page, it's not specific to 茨城. To me, this type of sentence-end だか is stereotypical いなか言葉 that is associated with no particular region in Japan. (That is not to say だか is an imaginary thing.)
It mostly corresponds to う・よう in Tokyo dialect (≒ べし in classical).
Summing up the sources below, it means 推量・勧誘・断定.
(推量) ～だろう / でしょう ⇒ ～べ (男性)
寒いだろう ⇒ 寒いべ
日本人でしょう ⇒ 日本人だべ
[意向形] / [辞書形]ぞ ⇒ [辞書形]べ (男性)
飲もう ⇒ 飲むべ
帰ろう ⇒ 帰るべ
Short answer is no, AFAIK.
However, if I may go beyond your restriction of "na-adjectives", things get interesting.
For example, daikon だいこん is pronounced as deekon でーこん. Daikon is not an adjective, but you see a similar pattern as sugoi / sugee.
BTW, this is not limited to Tokyo-ben. We say Okayama-ben turns "daikon taitoite" だいこんたいといて （...