Why would the RPMs on our car decrease and the car will sort of surge when we are coming to a atop

We recently changed the alternator in our car and we thought we fixed our car, however it started acting up the next day and within a few days our car wouldn't start at all. We took the battery and got it charged because they said it was almost dead. We thought we found the problem because as soon as the recharged battery was put back in the car, it started after 3 days (it had also been very rainy and cold and was warmer when the car started.) But when we drove the car we noticed that when we come to a stop the rmps go down and the car surges. We have asked several friends and we get a different answer each time, and I know there are A LOT of things that could be wrong but I dont have time or MONEY to go on a hunch and keep fixing things they think may be wrong. Some say it could be that our bad alternator killed a cell or two in the battery which makes it surge or "short circut' as he put it, so a new battery may solve our problem. Another said our fuel pump went out, but if that was the case why would my car have started when the battery was charged, if it was already broken down the fuel pump would have went out completly making it no driveable til replaced. Others say we may have gotten water in our fuel or bad gas that made our car run harder depleted the battery so we rain the gas out and put new in which made it run a little better but it still surges a little. Could it be a clogged fuel filter or something remotely simple? I appreciate your time and any advice will help!