Focusrite is probably not going to cut it honestly.
The best budget A/D's are the Audients. Really nice convertors in those.
What I would suggest is an active DI like the Radial J48. Get a used one for $250. It will make a huge difference.
I know exactly why your sound is not great right now, and can picture exactly what you are saying. One of the reasons is the signal needs a boost since it is passive, and the radial will do just that.
If you want to graduate to 3D pro tone, you will need to invest more and get a Reddi/Noble/Etc.. But you can do a lot with an active DI.
If the Radial is too much, grab a used Countryman. The main thing to look for in a DI on a passive bass (ordered by cost):
1. Output - you need a higher output. Solved with a basic active DI (start here)
2. Transformer - for the next step up you want a Jensen transformer. It does make a difference in how transients are handled. It will eliminate some of the harshness and cheap sound that you hear going straight in to inexpensive A/D's. A Radial JDI is a great option here, but it is a passive DI, and I still rank getting an active one higher as a priority.
3. Tubes - next level is tubes. When you get a DI with active output, a high quality transformer, plus the proper ability to drive tubes, you are now living the tone recording dream. It is the perfect combo of higher output (cleaner signal), higher quality transformers (better tracking of transients) and 3d tone (tubes adding upper harmonics to the sound). The bad news is that this combo is pretty expensive. Once you get serious, you will want this as your bass pre, and then your A/D will be a whole other story that I won't get into right now