PA-23 Pilot reported an engine lost a significant amount of oil which resulted in a diversion to landing. Maintenance determined a loose screw on a newly installed part caused the leak. After repairs were made the pilot departed without refueling; which resulted in fuel starvation and a diversion to landing.

Narrative

Landed at ZZZZ1 on Date. Approximately 110 NM from airport lost right engine vacuum pump. Left engine vacuum pump was working perfectly. In the ensuing days; I travelled to get a replacement vacuum pump. The same was installed at ZZZZ1 and upon starting right engine; vacuum pump operated normally.On Date1 my passenger and I departed ZZZZ1 on an IFR plan to ZZZ. Total fuel was 6 hours and 30 minutes. Estimated total time on route was 4 hours and 50 minutes. About two hours into the flight or 50 NM west of ZZZ we noticed significant amount of oil coming from inside the right engine. I contacted ZZZ Center; requested a divert to ZZZZ and received priority. Right engine was shut off as per airplane manual. Oil pressure and temperature never reached red mark. Landed uneventfully at ZZZZ and taxied to the FBO ramp.After required paper work was filled out and COVID-19 regulations were explained to us; including the possibility of mandatory quarantine for 7 days in a hotel room; despite the fact that both of us were vaccinated against it; we were serviced by an airplane mechanic who found out one of the nuts on the recently installed vacuum pump was lose and causing the oil leak. In order to tighten it up; he had to remove the right magneto. I called the A&P at ZZZZ and he confirmed he did not remove the magneto because he had a 'special tool' to tighten it up.After testing the airplane; we were able to depart on an IFR flight plan to our destination ZZZ at around XA:40 local time. Needless to say; in the duress of the moment I thought I had enough fuel to reach my destination plus 90 minutes reserve. Unfortunately; I neglected to deduct fuel consumption for a climb to 12;000 ft. While being vectored over ZZZ2 to ZZZ; both engines began to run rough despite fuel pumps on and switching tanks. Another [priority was received] and we landed uneventfully at ZZZ2. Next morning; I found out the airplane only had 16 gallons left on all four tanks. In the stress and haste of the moment; my mistake was not refueling at ZZZZ. I am sorry for the delays to air traffic this event caused.