The effect of variable dust charge, dust temperature and trapped electrons on small-amplitude dust acoustic waves is investigated. It is found that both compressive and rarefractive solitons as well as double layers exist depending on the non-isothermality parameter. A modified Korteweg de Vries (MKdV) equation is derived. Critical cases, at which the nonlinear coefficient is approximately zero, are derived. In the vicinity of the critical values, KdV and further MKdV (FMKdV) equations are obtained. Employing quasipotential analysis, the Sagdeev potential equation has been derived. Because of the presence of free and trapped electrons, the plasma acoustic wave exhibits different features of various solitary waves. The Sagdeev potential equation, at a small amplitude, shows that the ordering of non-isothermality in the dusty plasma plays a unique role. In the case of a plasma with first-order non-isothermality, the Sagdeev potential equation shows the compressive solitary-wave propagation while, for a plasma with higher-order non-isothermality, the solution of this equation reveals the coexistence of compressive and rarefractive solitary waves. In addition, for certain plasma parameters, the solitary wave disappears and a double layer is expected. Again, with the better approximation in the Sagdeev potential equation, more features of solitary waves, known as spiky and explosive, along with the double layers, are also highlighted. The findings of this investigation may be useful in understanding laboratory plasma phenomena and astrophysical situations.