As you mentioned, grid independence solution check 'must' be achieved prior to any FEA/CFD post-processing. The general trend is to reduce mesh density, more close to the region of interest and less globally, otherwise you have to pay the expensive computational cost, which I guess you reached to that point. In SW Sim FEA you can take advantage of mesh controls to increase mesh density. While doing this, you should pick at least one monitoring parameter in your analysis in order to monitor its deviation compared to previous mesh resolution case. For general mechanical engineering purposes, this deviation should become less than 'about' 8%; for other sensitive applications such as aerospace/military a lower margin is desirable. For FEA analysis, you should also be careful about 'stress singularity', wherein a high stress value appears at a sharp re-entrant and by increasing mesh density in that area it escalates considerably, marches towards infinity and will never converge. If that's the case, you need to get rid of singularity at the beginning using various techniques such as adding a small fillet etc or in worst case scenario, ignore the stress results close to that region.
Hope it helps,