An organization can be simultaneously a 501(c)3 and a 501(c)4 - often this is managed by having two separate legal entities with a mutual relationship, but it's possible for the same organization to be classified both ways.
Separately, a 501(c)3 is allowed to engage in lobbying - they are just subject to restrictions on how much time and money than can spend (last I checked, it's up to 10% of time and 10% of money).
This is all correct -- Notably, you can't spend 501(c)(3) dollars on (c)(4) activities. All of those employees that work for both entities have to accurately record their time and expenses the (c)(3) must charge the (c)(4) for any support that they give it. It's a big pain in the ass but worth it to keep the IRS off your case.
Separately, a 501(c)3 is allowed to engage in lobbying - they are just subject to restrictions on how much time and money than can spend (last I checked, it's up to 10% of time and 10% of money).