I have a bath lavatory with two sinks that drain into a common drain. I recently unclogged the drain and replaced the pop-up under one of the sinks. Since doing this, whenever I run water in one sink, I get bubbling of water in the pipe below the other sink, and at times, water backs up in the other sink. Both sinks are draining slowly. Does this mean I still have a clog deeper in the drain? Does the height of the traps or pop-ups for both sinks need to match for proper drainage, or should they drain ok if the pop-up height is off by a few inches?
The problem sounds like a blockage in the drain at some point, you also could have a blockage in the vent. All drain lines must have a vent to let the air escape, the bubbling in the opposite sink indicates air is trying to escape and the vent is not allowing it to exit trough the normal path.
Use a snake to clear the drain, remove the p-trap and run the snake as far down the drain line as you can. You will have to use a 1/4 inch or 3/8 inch snake on the drain.
To clear the vent you will have to snake the line from above. If you have a two story house have a plumber to the job, if you feel you can handle the job, rent a powered snake at your local Home Depot and clear the vent line. While you're up on the roof, clear all of the vents while you have a snake. A little preventive maintenance goes a long way.
If this doesn't solve the problem, you'll have to have a plumber check it out further.
I would agree with Mike, you have a clog further down the line; perhaps as a result of clearing the first clog.
If you're looking at a budget approach to small line drain cleaning I've had pretty good luck with an inexpensive snake and a cordless drill. You'll also need a helper. Pick up one of these snakes at your local Home Depot ($7.50). Brasscraft model BC91015, SKU: 167-436
You'll need to disassemble the traps and drain lines back to the wall because as the drain enters the wall the pipe will make a 90 degree turn downwards. You'll want as clean of a shot at that as you can get.
Chuck the plain end of the snake in a cordless drill and if the drill has dual speeds, set it on the lower speed. You'll have to stretch the snake out a bit because it will want to flop around. Put on a pair of work gloves and slowly feed the snake into the pipe at the same have whomever is running the drill to turn it on slowly. We're talking less than 75 rpm. The spinning of the snake will actually help it "sneak" around the curve in the pipe. You may have to try a few times, maybe from slightly different angles. Once you have it around the bend your helper can up the speed snake slightly as you work it and out of the pipe. Feed it in a foot and pull it 6", 2 feet and 12", and keep going like that. After a few feet, pull it out completely and see if there is anything on the end of the snake. If it looks clean, stick it back in and keep going. Sooner or later, I don't know when you're going to hit another turn or run out of snake. Reassemble the drain and test.
"Does this mean I still have a clog deeper in the drain?"
Yes. and the clog must be at or after the point where the 2 drains connect into one.
"Does the height of the traps or pop-ups for both sinks need to match for proper drainage, or should they drain ok if the pop-up height is off by a few inches?"
This should not be critical, unless your new pop-up dropped the p-trap down to the point that you had to finagle a new and strange drain set-up. Normally, you would trim the bottom of the new drain assembly so that it matched the length of the old one. The way the sink drains should be set up is to have the "P" part of the p-trap sit just below the level of the wall drain so that it always has water in it. The whole point is to prevent sewer gas from coming out of your sink drain, and the water in the trap prevents that.
In the picture on the left, you see the pipe leading to the wall drain on the left, and it is horizontal, just like the pipe on the 2nd picture that leads off to the right.
To the extent that one drain is lower than the other just means that the lower one will have more water in it, making it more likely to clog. In any event, you will need to disassemble the traps and clean them out. Without a picture of your sink drains it is hard to tell what else may be going on here.
You may find that the horizontal pipe leading out to the wall drain is where your main clog is. Since you have taken the pipes out, they become very easy to clean and replace. Do yourself a favor and buy new sealing washers for the drain nuts. If the pipes you have removed are not clogged, then you will have to snake out the wall drain. If these pipes are clogged, just take a good look at the wall fitting and clean it out right there. Note that this wall fitting becomes a "T" just inside the wall, and water drains straight down from there, an unlikely place for more clogging until well down the drain when it turns again.
Adam gave you the techniques for snaking out a drain, but I would much prefer to use a 1-man auger and pay the extra dollars for this tool:
You can hook up a drill to the back of this tool as well, and one person can easily handle it. The trigger on the front lets you automatically feed and retract the cable as you turn the drum.
Problems with sinks can be extremely annoying. Even a simple sink clog can cause serious flooding, which can damage drywall. If the DIY techniques don't work, it is advisable to call the plumbing experts. A qualified professional plumber like from drain cleaning company NJ can unclog your drains, and repair any plumbing issues, very quickly. Preventative maintenance is even more effective; regular drain cleaning in homes can stop these drains from clogging, and eventually flooding homes. For the best prevention homeowners should have their bathtub, shower, and sink drains cleaned by a professional once each year.