What you are saying is, unfortunately, not true. Under the NLRA, Unions are required to represent the non-dues paying workers just as if they were members. Also note that they get the full benefit of the contracted wages, benefits, and the right to have a union representative at any discipline hearing.
Case law, from the AFL-CIO's website:
A private sector union operates under a legally enforceable "duty of fair representation," that is, the union must “fairly and equitably…represent all employees..., union and nonunion.” International Assn. of Machinists v. Street, 367 U.S. 740, 761 (1961). This means a union cannot discriminate or act arbitrarily toward any employee due to the nature of his relationship with the union, and all employees are equally entitled to the union’s fair and vigorous representation. All members and non-members must receive the fruits of the union’s bargaining – wages, benefits and all other rights and protections – and enjoy full access to the grievance and arbitration process that is established to redress adverse or improper actions by the employer. 29 U.S.C. § 158(b)(2); Steele v. Louisville & Nashville R.R. Co., 323 U.S. 192, 204 (1944); Bowman v. Tennessee Valley Authority, 744 F. 2d 1207, 1213-14 (6th Cir. 1984). This right to full and fair individual treatment by the union is legally enforceable in court and before the NLRB. Vaca v. Sipes, 386 U.S.171 (1967); Plumbers Local 32 v. NLRB, 50 F. 3d 29, 31-32 (D.C. Cir.), cert. denied, 516 U.S. 974 (1995).