I develop a new analysis of English “weak necessity” modals 'should' and 'supposed to' that is motivated by novel observations of extra-weak meanings in some environments. For example, *'supposed to'* in (1a) gives rise to a meaning that can be paraphrased as the weaker modal statement in (1b). (1) A: Can you please pick up a book for me at the office? 1. B: I don’t know... Are we supposed to be back on campus without a Covid test? 2. B′: I don’t know... Is it okay for us to be back on campus without a Covid test? I argue that this evidence of weak readings suggests that these modal verbs exhibit a type of variable force. Sentences with weak necessity modals express universal force in positive sentences and existential force under negation. The analysis will build on an analogy with free-choice disjunction that assumes a basic weak meaning that strengthens in upward-entailing environments (Fox, 2007; Bassi and Bar-Lev, 2016). These modals' "neg-raising" property then follows as an immediate consequence. I further propose that the precise distribution of the strengthened readings is governed by the polarity-sensitive nature of the modals (cf. Iatridou and Zeijlstra, 2013; Homer, 2015). In particular, I argue that the polarity sensitivity of the modals is the result of the association of their domains with a covert 'even' (cf. Lahiri, 1995; Crnicˇ, 2014, 2019 for NPI 'any'). I then show how rhetorical effects observed in questions like (1a) can be generated with independently-motivated mechanisms proposed in Iatridou and Tatevosov (2016) to explain the effects of 'even' in questions.