Security is always about creating a compromise between security and functionality. Which means that to make the best security for you, you need to consider what are your essential functions, what are your assets, and what are your threats.
The government doesn't give a crap about getting into your bank or finding out about your affair. If those are your concerns, then government funded tools are great. While they may be more likely to have government back doors in them (I'll address that in a bit) they are much, much less likely to have accidental vulnerabilities or to be compromised by a third party. The government cares a lot about security, and has no shareholders to answer to about why they are spending so much money securing trivial software.
On the other hand, random commercial or open source software may not have back doors built in to them, but may instead have a laundry list of vulnerabilities that leave you at least as exposed. Open source is only as secure as volunteers care to make it, and commercial software is only as secure as the development team can justify to the board.
If the activity you are trying to secure is illegal, then you need to take a closer look at things. In particular, you should look at which agency. TOR, for instance, is not supported by the FBI as you claim. In fact, the FBI hates it and actively works to undermine it by targeting end nodes. It's supported by the State Department. Why does that matter? Because the State Department's mission is to maintain international relations and to prevent the outbreak of (unintended) war. They don't give a crap about your little drug ring or even your child pornography distribution ring. Their concern is that if someone eavesdrops on their communications, it might lead to a war in which thousands of people are killed. They are not willing to compromise the security of nations in order to help the FBI prosecute some small potatoes crook. So while the security of TOR has some concerns, it's not because of the State Department's involvement.
You need to evaluate your options against your own needs. And you need to do so dispassionately. The way you lump together "government" and the FBI suggests to me you are approaching this from a predetermined conclusion. Ultimately, one of the principal tenets of security is that you can't entirely trust anything. If you are relying on one specific tool or technology to keep you safe, you aren't.