Hello OP
The behaviour you describe is exactly how my ex-husband started to behave. He'd go through my bag (I would leave it in the kitchen at night), my pockets, my internet history.
He had convinced himself that he just wasn't snooping hard enough and it escalated over a period of 3 years. He recorded me in the house - video and audio - and had me followed by his mates. I would be given an allotted amount of time to get the weekly shopping and would be questioned at length if I took longer than expected (which shops did I go to, who did I talk to, etc etc). When my train home was delayed, I would have to text him a picture of the departure board to prove that I hadn't gone out after work.
He isolated me from my friends and family and told what I could and could not wear to work.
Most scary of all, he stole my used underwear and said he was going to send it away for forensic analysis. (He was high-ranking in the police at the time). He became very emotionally and physically abusive and ended up having a full-on psychotic episode. He got a 12-week suspended sentence for one thing he did to me - by then I was well aware that I couldn't 'fix' him. He only left me alone because of a restraining order.
What my ex did was so slow and insidious at the start that I didn't see how dangerous he was becoming.
I hear through extended family members that he still wonders how I managed to 'get away with it' when the truth of the matter is that I am a very boring, plain-living, full-time working mum with never the inclination nor time to be a cheat. Ironically HE was the cheat (three women, to my knowledge).
I'm not saying that your husband will go to these extremes, however I am concerned by his inability to acknowledge that what he is doing is exceptionally disrespectful to you. Like my ex, he seems to think he just hasn't found the evidence yet and cannot accept that there is no evidence to find.
If your husband is anxious about other aspects of life I would suggest he visits his GP.
Are you happy to go on living with him if he refuses to stop snooping, or is this a deal-breaker for you? FWIW I don't think he'll stop snooping unless you issue an ultimatum and are willing to see it through.
Good luck 