Yes, both Asana and Timesheet offer a Free Trial.
Asana is designed for Large Enterprises, Medium Business and Small Business.
Timesheet is designed for Individuals, Freelancers, Medium Business and Small Business.
Pricing for Asana Starts at $6.92/user/month when monthly whereas for Timesheet Starts at $5/Month.
Some top alternatives to Asana includes Quickbase , Smartsheet, ClickUp, Zoho Projects, Jira, Basecamp, Trello, Wrike, ProofHub, monday.com, Teamwork, ProjectManager.com, Microsoft Project, Quick Base and Height.
Some top alternatives to Timesheet includes Time Doctor, My Hours, TrackingTime, Clockify, TimeTrack, Timebase, Time Track Bill, Time Analytics, Billable and Time Tracker by eBillity.