"Ich stand unter Zugzwang" is just as correct as "Das neue Produkt der Konkurrenz bringt uns in Zugzwang", although being "under" Zugzwang sounds more common to me. You could also say "Das Produkt sorgt für Zugzwang".
"Unter Zugzwang stehen" and "in Zugzwang bringen" both sound natural to me, but "für Zugzwang sorgen" doesn't. (It's not incorrect, but it's not a common combination.)
