As you can read in the release notes of the latest dbSNP build, they have mapped their variants both to GRCh38 (hg20) and GRCh37(hg19):
dbSNP has released human Build 141 based on the GRCh38 assembly (http://www.ncbi.nlm.nih.gov/assembly/GCF_000001405.26/), as well as on the GRCh37.p13 assembly.
If you go for example to the dbSNP page for rs123, you can see that in the section "Integrated Maps" the genomic position in both assemblies is displayed, 7:24926827 in GRCh38 and 7:24966446 in GRCh37.13.
However, I don't think it's possible to search dbSNP with GRCH37 based positions ....
I would suggest to use the Variant Effect Predictor (VEP) (which is still GRCh37 based) to investigate if there are already known variants for the same genomic positions of your potential variants.
Hope this helps.
PS - The official names of the assemblies are GRCh37 and GRCh38, so please use these instead of hg19 and hg20.