 Re: sincerity to help
Both wrong and unnatural.
a) Wrong because sincerity is not followed by an infinitive but by the preposition in - so any verb will be in the gerund. Here are some examples from a concordancer...
010. ...be expected to put much emphasis on this as evidence of its SINCERITY in "resisting" the wage pressures of a powerful ...
011. ... beneficently placing it in her hands. "As a pledge of my SINCERITY in the matter of the crowning of Prince Richard, ...
015. make the central proposal; but the East would have to show SINCERITY in carrying it out". "But where is the leader who ...
You could therefore change your sentence to : I appreciate your sincerity in wanting/offering to help
b) Unnatural because it's fairly rare to talk about sincerity "in doing something". Generally it's just used to describe an abstract quality of the person in terms of the specific context - so if anything follows at all, it's more likely to be of + noun phrase . Some more examples :
018. The unionist representatives continue to doubt the SINCERITY of the IRA cease-fire and have called on Sinn Fein...
019. it so!" "You are very good to me, my lady," Joan said with SINCERITY. "I know not how I may repay you." "I hold you in
020. "I know moreover that you speak with SINCERITY. Tell me then, sir, if you yourself believe that n
021. ... met his gaze. There could be no doubt of the youth's SINCERITY.
Another phrasing could therefore be : I appreciate the sincerity of your offer to help |