N3 — outside your level filter
but; and yet — the plain-style written "however"
何度も彼に電話をかけた。だが、一度も出なかった。
nando mo kare ni denwa o kaketa. daga, ichido mo denakatta.
I called him again and again. But he never once picked up. neutral
計画そのものは悪くない。だが、今の予算では実現できない。
keikaku sonomono wa warukunai. daga, ima no yosan de wa jitsugen dekinai.
The plan itself isn't bad. Yet with the budget as it stands it can't be carried out. neutral
Variants
ですがdesugaformal
the polite face — meetings, service speech and mail close to speech; formal ですます writing still opens with しかし far more often.
だがしかしdagashikashicolloquial
doubles up for effect; commentary, blogs and speech reaching for drama, never a report.
だが
だがdagaN3formal