I picked up this card at Best Buy: Newegg.ca - PNY VCG841024D3SXPB GeForce 8400 GS 1GB 64-bit DDR3 PCI Express 2.0 x16 HDCP Ready Video Card
WEI dropped to 4.6/5.4 for Graphics/gaming graphics.
No stuttering on a 1080p .mp4 and .mkv, both of which my G620 stuttered on.
However, I ended up returning the card and purchased an i3-2105. I believe this should be sufficient.
Originally I intended to return the card and sell my Pentium, which would be enough to pay for the Pentiun G850 (which according to Newegg has HD 2000), but it didn't have HD 2000, so I looked at the i3-2100 which was $120, and the i3-2105 with HD 3000 was only $30 more.
Theoretically I should be happy with this setup for the time being